Highlights

  • Neural computation relies on compartmentalized dendrites to discern inputs

  • A method is described to systematically derive the degree of compartmentalization

  • There are substantially fewer functional compartments than dendritic branches

  • Compartmentalization is dynamic and can be tuned by synaptic inputs

Summary

The dendritic tree of neurons plays an important role in information processing in the brain. While it is thought that dendrites require independent subunits to perform most of their computations, it is still not understood how they compartmentalize into functional subunits. Here, we show how these subunits can be deduced from the properties of dendrites. We devised a formalism that links the dendritic arborization to an impedance-based tree graph and show how the topology of this graph reveals independent subunits. This analysis reveals that cooperativity between synapses decreases slowly with increasing electrical separation and thus that few independent subunits coexist. We nevertheless find that balanced inputs or shunting inhibition can modify this topology and increase the number and size of the subunits in a context-dependent manner. We also find that this dynamic recompartmentalization can enable branch-specific learning of stimulus features. Analysis of dendritic patch-clamp recording experiments confirmed our theoretical predictions.

Graphical Abstract

Figure thumbnail fx1 {focus_keyword} Electrical Compartmentalization in Neurons

Keywords

  • neural computation
  • dendrites
  • compartmentalization
  • independent subunits
  • branch-specific learning
  • dendritic computation

Introduction

Brain function emerges from the orchestrated behavior of billions of individual neurons that transform electrical inputs into action potential (AP) output. This transformation starts on the dendritic tree, where inputs are collected, and proceeds to the axon initial segment where APs are generated before they are transmitted to downstream neurons through the axon. While axons appear to merely communicate the neuronal output downstream, dendrites collect and nonlinearly transform the input. This phenomenon, termed dendritic computation, has been shown to occur in vivo and to be required for normal brain function (

). In both experimental and theoretical work, an abundance of dendritic computations have been proposed (

,

,

). Nearly all of them assume that dendrites are compartmentalized into independent subunits: regions on the dendritic tree that can integrate inputs independently from other regions.

The computational significance of these subunits arises from their ability to support independent regenerative events, such as N-methyl D-aspartate (NMDA), Ca2+, or Na+ spikes (

Wei et al., 2001

  • Wei D.-S.
  • Mei Y.-A.
  • Bagal A.
  • Kao J.P.Y.
  • Thompson S.M.
  • Tang C.-M.

Compartmentalized and binary behavior of terminal dendrites in hippocampal pyramidal neurons.

). These events, where an initial depolarization is enhanced supralinearly by subsequent synaptic inputs and/or voltage-dependent ion-channel currents (

Major et al., 2013

  • Major G.
  • Larkum M.E.
  • Schiller J.

Active properties of neocortical pyramidal neuron dendrites.

), significantly strengthen the computational power of neurons. They enable the local decoding of bursts of inputs (

Polsky et al., 2009

  • Polsky A.
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  • Schiller J.

Encoding and decoding bursts by NMDA spikes in basal dendrites of layer 5 pyramidal neurons.

) and hence, through branch-specific plasticity (

Golding et al., 2002

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Dendritic spikes as a mechanism for cooperative long-term potentiation.

,

Govindarajan et al., 2011

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  • Israely I.
  • Huang S.-Y.
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The dendritic branch is the preferred integrative unit for protein synthesis-dependent LTP.

,

Losonczy et al., 2008

  • Losonczy A.
  • Makara J.K.
  • Magee J.C.

Compartmentalized dendritic plasticity and input feature storage in neurons.

,

Weber et al., 2016

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  • Polito M.
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Location-dependent synaptic plasticity rules by dendritic spine cooperativity.

), drive the clustering of correlated synaptic inputs (

Gökçe et al., 2016

  • Gökçe O.
  • Bonhoeffer T.
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Clusters of synaptic inputs on dendrites of layer 5 pyramidal cells in mouse visual cortex.

,

,

Lee et al., 2016

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  • Reed M.
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Anatomy and function of an excitatory network in the visual cortex.

). A recent finding that distal apical dendrites can spike 10-fold more often than somata (

Moore et al., 2017

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Dynamics of cortical dendritic membrane potential and spikes in freely behaving rats.

) suggests an important role for this branch-specific plasticity. Independent subunits furthermore allow different input streams to be discriminated from each other (

Johenning et al., 2009

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Dendritic compartment and neuronal output mode determine pathway-specific long-term potentiation in the piriform cortex.

), and they facilitate sensory perception through feedback signals (

Takahashi et al., 2016

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Active cortical dendrites modulate perception.

).

When triggered independently, these local regenerative events are predicted to enable individual neurons to function as two-layer neural networks (

,

Poirazi et al., 2003b

  • Poirazi P.
  • Brannon T.
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Arithmetic of subthreshold synaptic summation in a model CA1 pyramidal cell.

). This in turn should enable neurons to learn linearly nonseparable functions (

Schiess et al., 2016

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Somato-dendritic synaptic plasticity and error-backpropagation in active dendrites.

) and implement translation invariance (

Mel et al., 1998

  • Mel B.W.
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Translation-invariant orientation tuning in visual “complex” cells could derive from intradendritic computations.

). On the network level, independent subunits are thought to dramatically increase memory capacity (

Poirazi and Mel, 2001

  • Poirazi P.
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Impact of active dendrites and structural plasticity on the memory capacity of neural tissue.

), to allow for the stable storage of feature associations (

Bono and Clopath, 2017

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Modeling somatic and dendritic spike mediated plasticity at the single neuron and network level.

), represent a powerful mechanism for coincidence detection (

Chua and Morrison, 2016

  • Chua Y.
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Effects of calcium spikes in the layer 5 pyramidal neuron on coincidence detection and activity propagation.

,

Larkum et al., 1999

  • Larkum M.E.
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A new cellular mechanism for coupling inputs arriving at different cortical layers.

), and support the back-prop algorithm to train neural networks (

Guerguiev et al., 2017

  • Guerguiev J.
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Towards deep learning with segregated dendrites.

,

Sacramento et al., 2017

Sacramento, J., Costa, R.P., Bengio, Y., and Senn, W. (2017). Dendritic error backpropagation in deep cortical microcircuits. arXiv, arXiv:1801.00062v1.

,

).

Thus, abundant data show that dendritic trees consist of a multitude of subunits, and both experimental and theoretical work suggests an important computational role for these subunits (

Major et al., 2013

  • Major G.
  • Larkum M.E.
  • Schiller J.

Active properties of neocortical pyramidal neuron dendrites.

). Nevertheless, to date, there is no criterion to quantify the conditions under which regions on the dendrite support the independent triggering of regenerative events, and there is no clear idea about the number of such subunits that can coexist on a given dendritic tree. The most prevalent hypothesis likens dendritic subunits to individual branches (

Branco and Häusser, 2010

  • Branco T.
  • Häusser M.

The single dendritic branch as a fundamental functional unit in the nervous system.

).

In this work, we develop a computational method to answer these questions. We link the dendritic arborization to an impedance-based tree graph and show how the topology of this tree graph reveals independent subunits. We find that even though voltage may decay sharply toward branchpoints, the remaining depolarization can still cause cooperative dynamics, hence limiting the number of compartments that coexist on dendritic trees. We show that the degree of cooperativity between synapses can be summarized with a single, measurable number, which we term the impedance-based independence index Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. We compute this index from dendritic patch-clamp recordings from basal dendrites of layer 5 (L5) pyramidal neurons in the somatosensory cortex (

Nevian et al., 2007

  • Nevian T.
  • Larkum M.E.
  • Polsky A.
  • Schiller J.

Properties of basal dendrites of layer 5 pyramidal neurons: a direct patch-clamp recording study.

) and find the obtained values to agree with our models. We finally demonstrate that compartmentalization is not a static concept but can be regulated dynamically, since balanced inputs or shunting inhibition can modify the topology of the impedance-based tree graph and increase compartmentalization in a global or a highly local manner. This increase in compartmentalization can decorrelate synaptic weight dynamics and hence enable the branch-specific learning of stimulus features.

Results

 A Drastic Simplification of the “Connectivity Matrix” of a Neuron

Formalizing the concept of a dendritic subunit requires studying the generation of local regenerative events in conjunction with the electrical separation between synapses. Consider the following scenario: if we were to depolarize a dendritic branch by an amount Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (for instance, by injecting a current), then we would alter the probability for local synapses to elicit regenerative events (for instance, through the voltage dependence of synaptic currents such as the NMDA current;

Jahr and Stevens, 1990a

  • Jahr C.E.
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A quantitative description of NMDA receptor-channel kinetic behavior.

,

MacDonald and Wojtowicz, 1982

  • MacDonald J.F.
  • Wojtowicz J.M.

The effects of L-glutamate and its analogues upon the membrane conductance of central murine neurones in culture.

). If Math Eq {focus_keyword} Electrical Compartmentalization in Neurons were small enough, then the change in this probability would be negligible, and the local synapses could be considered independent with respect to the perturbation. In the brain, these changes in voltage are caused by synapses elsewhere in the dendritic tree, which may also elicit regenerative events themselves. Thus, we must ask whether regenerative events elicited elsewhere are attenuated to a sufficient level, allowing local synapses to elicit regenerative events independently.

However, under conditions in which many synapses are active, it is difficult to untangle the loci of origin of the fluctuations at any one dendritic site. Fluctuations that facilitated local regenerative events may have originated in nearby sibling branches, more centripetal or centrifugal branches, or wholly different subtrees. Additionally, the degree to which these fluctuations propagate to any other locus on the neuron is not fixed but depends on all other inputs through synaptic shunting (

Gidon and Segev; 2012

  • Gidon A.
  • Segev I.

Principles governing the operation of synaptic inhibition in dendrites.

). Thus, fluctuations at any one site are a tangled, context-dependent combination of the inputs at all other sites. Here, we solve this “tangling” problem by expressing the voltage in a way that these fluctuations are never tangled in the first place: we express the dendritic voltage as a superposition of voltage components at different spatial scales (Figure 1A).

Figure thumbnail gr1 {focus_keyword} Electrical Compartmentalization in Neurons

Figure 1Construction of the NET

Show full caption

(A) Schematic depiction of the NET. The dendrite voltage is obtained as a superposition of voltage components (left) at different spatial scales. The structure of this superposition can be represented graphically as a tree graph (right).

(B) Average impedance kernels for impedances within the indicated ranges on a granule cell.

(C) Granule cell dendrite with three inputs regions (numbers 1–3) and somatic readout.

(D) Impedance matrix associated with the granule cell (morphology is unraveled in a depth-first manner [

]; see also Figures S1A–S1C). Color encodes the impedance between each pair of points.

(E) Points in the impedance matrix of the granule cell where the impedance is between Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (i.e., the minimal transfer impedance in the cell) and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons are colored blue. The average of the impedance kernels associated with these points will constitute the impedance kernel of the NET root.

(F) Input impedance (diagonal of the impedance matrix). Connected domains with input impedance larger than Math Eq {focus_keyword} Electrical Compartmentalization in Neurons are indicated on the x axis and denoted by Math Eq {focus_keyword} Electrical Compartmentalization in Neurons; they will constitute the child nodes of the NET root.

(G) Impedance matrix with dendritic domains associated with each child node Math Eq {focus_keyword} Electrical Compartmentalization in Neurons marked by a red square. These restricted matrices are used in the next step of the recursive algorithm.

(H) Full NET for the granule cell. Nodes where inputs from regions 1 to 3 arrive are indicated in the corresponding color.

(I) Pruned NET obtained by removing all nodes that do not integrate regions 1–3.

(J) NET voltages components after an input to location 1 (left), 2 (middle), and 3 (right) (onset indicated with a vertical line, its color matching the input location in (C)). Leaf voltage components Math Eq {focus_keyword} Electrical Compartmentalization in Neurons are unique to the associated regions while other nodal voltage components (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) are shared between regions.

(K) Dendritic voltages for the soma and input regions (Math Eq {focus_keyword} Electrical Compartmentalization in NeuronsMath Eq {focus_keyword} Electrical Compartmentalization in Neurons), obtained by summing the nodal voltage components on the path from root to associated leaf. NET traces (dashed lines) agrees with the equivalent NEURON model (full lines).

Our model of superimposed voltage components can be visualized as a tree graph, which we term the neural evaluation tree (NET), where each node represents such a component and integrates only inputs arriving into its subtree (Figure 1A, right). The root node of the tree (yellow node in Figure 1A) integrates all inputs to the neuron, whereas the leaf nodes (pink and blue nodes in Figure 1A) only integrate inputs in parts of local subtrees or branches. In this tree, the locus of origin of fluctuations is readily available: if fluctuations that influence regenerative events at synapses 1 and 2 originated at synapse 3, then they would be visible in the voltage component of the green node in Figure 1A. Thus, if the green node contributed to regenerative events at synapses 1 and 2, then these synapses could not be independent from synapse 3. Conversely, if voltage fluctuations in the yellow node do not contribute to regenerative events at synapses 1 and 2, then these synapses must be independent from synapses 4 and 5.

Both the shape and the electrical properties of the dendritic tree determine the interplay between the voltage components; we must accurately estimate the voltage change following any one synaptic input throughout the whole dendritic tree. This suggests starting from the impedance matrix associated with a dendrite. This matrix (already hypothesized to be related to the dendritic compartmentalization;

Cuntz et al., 2010

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One rule to grow them all: a general theory of neuronal branching and its practical application.

) can be seen as the connectivity matrix of a neuron: Each element Math Eq {focus_keyword} Electrical Compartmentalization in Neurons describes how an input current at location Math Eq {focus_keyword} Electrical Compartmentalization in Neurons along the dendrite will change the voltage at any location Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. In prior work, the values Math Eq {focus_keyword} Electrical Compartmentalization in Neurons of this matrix (termed input impedances when Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and transfer impedances when Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) were used to shed light on the processing properties of various cell types (

Koch et al., 1982

  • Koch C.
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Retinal ganglion cells: a functional interpretation of dendritic morphology.

,

Koch et al., 1983

  • Koch C.
  • Poggio T.
  • Torre V.

Nonlinear interactions in a dendritic tree: localization, timing, and role in information processing.

) and interactions between synaptic conductances (

Gidon and Segev, 2012

  • Gidon A.
  • Segev I.

Principles governing the operation of synaptic inhibition in dendrites.

,

Koch et al., 1990

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Visibility of synaptically induced conductance changes: theory and simulations of anatomically characterized cortical pyramidal cells.

).

As we aim to approximate both the spatial and temporal neural dynamics, the impedance matrix is in fact a matrix of temporal kernels Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Each node must furthermore capture an average interaction between synapses; in Figure 1A, the pink node captures interactions between synapses 1 and 2, the green node between 1 and 3 and between 2 and 3, the blue node between 4 and 5, and the yellow node between 1 and 4, 1 and 5, 2 and 4, 2 and 5, 3 and 4, and 3 and 5. Thus, we construct an impedance kernel associated with a node as the average of the transfer impedance kernels between synapses for which that particular node captures the interactions. As we will see, the NET is constructed so that all interactions captured by a single node are approximately equal in amplitude. Furthermore, impedance kernels with equal steady-state impedance values share similar timescales (Figure 1B) so that averaging indeed yields accurate dynamics. While some variation is visible in these kernels on timescales smaller than 0.5 ms, the effect of these variations on the voltage dynamics is negligible, since input currents in general vary more slowly.

We illustrate our NET derivation algorithm on a granule cell (Figures 1C–1I). The dendritic impedance matrix, which is ordered by unraveling the morphology in a depth-first manner (

; Figures S1A–S1C), supports a tree-like description; a relatively even blue surface covers most of the matrix (Figure 1D), representing transfer impedances between main dendritic branches. Note that to a good approximation, these impedances indeed have similar magnitudes. Closer to the diagonal, squares of light blue are present, representing dendritic subtrees whose branches are electrically closer to each other than to other dendritic subtrees. On the diagonal, one finds small squares of green and red colors representing individual terminal branches. First, the impedance kernel of the global node (root node of the tree) is defined by averaging impedance kernels that relate inputs on different main branches (Figure 1E). Then, regions of the dendrite that are electrically closer to each other than to other regions are identified as connected regions where the input impedance is above the impedance associated with the global node. Due to the depth-first ordering, these regions show up as uninterrupted intervals on the diagonal of the impedance matrix (Figure 1F). New nodes are then added as child nodes of the root node (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons to Math Eq {focus_keyword} Electrical Compartmentalization in Neurons in Figure 1F), and each of these child nodes integrates regions from within one uninterrupted interval. Next, for each of the child nodes, the impedance matrix is restricted to its associated regions (which now correspond to the dendritic subtrees; red squares in Figure 1G), and the procedure is repeated until we reach the maximal value of the impedance matrix, since at that point, the whole dendrite is covered.

To study interactions between a subset of input regions on the dendritic tree, the full NET is pruned (see STAR Methods) so that only nodes that integrate the regions of interest are retained (Figures 1H and 1I). Note that we employ two ways of visualizing the NET. When we plot the full NET associated with a neuron, we collapse all regions in a section without bifurcations onto a single line, so that the graphical structure mimics the original morphology (Figure 1H). When we prune the NET so that the set of regions is sufficiently small, we plot the full tree (Figure 1I). The vertical lengths of the branches leading up to the nodes are plotted proportional to the impedance associated with that node. This gives a visualization of how electrically close regions on the morphology are (as in the morpho-electric transform by

Zador et al., 1995

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The morphoelectrotonic transform: a graphical approach to dendritic function.

).

We illustrate how the dendritic voltage at these regions of interest is constructed by the NET. Each nodal voltage component is computed by convolving the impedance kernel at that node with all inputs to its subtree (Figure 1J). The local dendritic voltage at a location is then constructed by summing the NET voltage components of all nodes on the path from root to leaf (Figure 1K), so that the local voltage is indeed a superposition of both global and local components. Note that this local voltage coincides with the voltage computed through simulating an equivalent neuron model (

) (dashed versus full line in Figure 1K). The key observation here is that the NET leafs (nodes Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) do not receive inputs from other locations (Figure 1J), in contrast to the local voltages in the biophysical model (Figure 1K). Furthermore, while in this example we have only implemented α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) synapses, it is straightforward to use other voltage dependent currents (i.e., NMDA or voltage-gated ion channels).

In conclusion, by formulating the NET framework, we have solved the tangling problem. We have introduced voltage components at the NET leafs that only depend on local inputs while being able to accurately model synaptic interactions. To assess independence, we only have to quantify the influence of global NET components on the voltage-dependent factors in the local synaptic currents. After validating the NET framework, we will turn our attention to this question.

 Validation of the NET Framework

We derived NETs for three vastly different exemplar morphologies (a cortical stellate cell [

Wang et al., 2002

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], a granule cell [

Carim-Todd et al., 2009

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], and a L5 cortical pyramidal cell [

Hay et al., 2011

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Models of neocortical layer 5b pyramidal cells capturing a wide range of dendritic and perisomatic active properties.

]; Figures 2A–2C). We then reconstructed the impedance matrix of the NET (Figures 2A–2C, top right panels) and compared it with the exact impedance matrix. The small value of the root mean square error (RMSE) between both matrices suggests that the NET will accurately reproduce the full neuronal dynamics (Figure 2D). To ascertain this, we implemented a simulation algorithm for the NET (see Methods S1), equipped each NET with somatic AP channels, and simulated it for 100 s while providing Poisson inputs to 100 randomly distributed excitatory and inhibitory synapses (see STAR Methods for all the simulation parameters). The resulting somatic voltage traces coincide with the traces obtained from equivalent NEURON simulations (

) (Figures 2A–2C, bottom right panels). Consequently, somatic RMSEs are low while spike prediction is excellent (Figure 2E). Thus, the RMSE between the true impedance matrix and its NET approximation is a good predictor of the accuracy of the NET. While here the synapse distribution was random, NETs are equally accurate when synapses are clustered (Figures S1D and S1E). Furthermore, intra-dendritic dynamics and regenerative NMDA events are well reproduced by NETs (Figures S1E and S1F).

Figure thumbnail gr2 {focus_keyword} Electrical Compartmentalization in Neurons

Figure 2Validation of the NET Framework and the Impedance-Based Independence Index Math Eq {focus_keyword} Electrical Compartmentalization in Neurons

Show full caption

(A) Cortical stellate cell.

(B) Hippocampal granule cell.

(C) Cortical L5 pyramidal cell. For each cell, morphology is shown on the left, the exact impedance matrix and its NET approximation on the top right, and the somatic voltage traces from equivalent NET and NEURON simulations on the bottom right.

(D) RMSE between NET and exact impedance matrices for each cell.

(E) Subthreshold somatic voltage RMSE (blue) and spike coincidence factor (red).

(F) Size of fluctuations of each nodal voltage component (quantified by its SD) as a function of the associated nodal impedance. The black line is a linear least-squares fit to the data.

(G) Voltage correlation between each pair of synapses on the three cells as a function of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Lines are obtained from kernel regression.

 A Single Number to Approximate the Degree of Independence between Regions

To speak of mutual independence between pairs of regions, their layout needs to be somewhat symmetric, so that Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons do not differ too much. If region 1 would, for instance, be much closer to a bifurcation than region 2, so that Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, then a regenerative event at region 1 would propagate without much attenuation to region 2. Hence, it would be impossible to independently generate local spikes at region 2. On the other hand, regenerative events at region 2 would substantially attenuate to region 1, so that it might be possible to have independent regenerative events at region 1. Thus, mutual independence and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons are only meaningful if the regions lie in a somewhat symmetric configuration around the nearest bifurcation.

 The Threshold for Independence

Correlations between regions on the dendrite decrease in a continuous fashion as a function of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (Figure 2G). We nevertheless wondered whether it is possible to find a threshold Math Eq {focus_keyword} Electrical Compartmentalization in Neurons above which regions constitute independent subunits (i.e., where local regenerative events are elicited independently from ongoing dynamics elsewhere).

Experimentally, the idea has emerged that every branch can be considered as an independent computational subunit (

Branco and Häusser, 2010

  • Branco T.
  • Häusser M.

The single dendritic branch as a fundamental functional unit in the nervous system.

). Sharp drops in transfer impedance across bifurcation points may lead to this idea, as delivering relatively few stimuli to a single dendritic branch (for instance, through two-photon glutamate uncaging;

Pettit et al., 1997

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) leads to a marked attenuation of the measured signal (typically Ca2+ dye luminescence or voltage) across bifurcation points (

Wei et al., 2001

  • Wei D.-S.
  • Mei Y.-A.
  • Bagal A.
  • Kao J.P.Y.
  • Thompson S.M.
  • Tang C.-M.

Compartmentalized and binary behavior of terminal dendrites in hippocampal pyramidal neurons.

) (Figure 3A). Nevertheless, inferring the extent of the dendritic subunits in this way is somewhat problematic. First, the Ca2+ signal itself is thought to be more local than the voltage signal (

Biess et al., 2011

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Barriers to diffusion in dendrites and estimation of calcium spread following synaptic inputs.

,

Nevian and Sakmann, 2004

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Single spine Ca2+ signals evoked by coincident EPSPs and backpropagating action potentials in spiny stellate cells of layer 4 in the juvenile rat somatosensory barrel cortex.

,

Simon and Llinás, 1985

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,

Yuste and Denk, 1995

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) while also being influenced by the nonlinear activation function of voltage-dependent calcium channels (

Almog and Korngreen, 2009

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Characterization of voltage-gated Ca(2+) conductances in layer 5 neocortical pyramidal neurons from rats.

). Second, even though there may be a relatively large amount of voltage attenuation across bifurcation points (Figures 3B and 3C), the remaining depolarization can still cause cooperativity between synapses during an ongoing barrage of inputs. To test this idea, we constructed a toy model with two leafs and one root representing, for instance, a single trunk that bifurcates into two child branches. This model allowed us to vary Math Eq {focus_keyword} Electrical Compartmentalization in Neurons on a log scale between 0.1 and 100 (Figure 3D, left) by changing the ratio of leaf versus root impedance while keeping the total input impedance at each site constant. We then stimulated synapse 1 with a supra-threshold conductance (strong enough to elicit an NMDA spike) and measured the depolarization in branch 1 and branch 2. It can be seen that there is substantial voltage attenuation; at Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, only 50% of the depolarization in branch 1 arrives at branch 2 (Figure 3D, right panel). Nevertheless, membrane voltage correlations decrease less as a function of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, so that for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, this correlation is still close to 1 (Figure 2G).

Figure thumbnail gr3 {focus_keyword} Electrical Compartmentalization in Neurons

Figure 3Voltage Attenuation Makes Dendritic Branches Look Highly Compartmentalized

Show full caption

(A) Example of a typical experimental situation. A branch is stimulated through two-photon glutamate uncaging (dashed circle, left), and a correlate of the voltage (here Ca2+, right) is measured (adapted with permission from

Wei et al., 2001

  • Wei D.-S.
  • Mei Y.-A.
  • Bagal A.
  • Kao J.P.Y.
  • Thompson S.M.
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Compartmentalized and binary behavior of terminal dendrites in hippocampal pyramidal neurons.

).

(B) Spatial voltage (computed with NEURON) in an oblique apical fork 10 ms after supra-threshold stimulation of an NMDA synapse at region 1. A rapid decrease of the depolarization makes the branches seem highly compartmentalized. Math Eq {focus_keyword} Electrical Compartmentalization in Neurons between regions 1 and 2.

(C) Trace of the depolarization at regions 1 and 2.

(D) Toy NET model, representing a dendritic fork, where Math Eq {focus_keyword} Electrical Compartmentalization in Neurons could be varied at will (left). Depolarization at regions 1 and 2 is shown as a function of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons following an input to region 1 (right).

We thus aim at constructing a criterion for independence between dendritic regions that is valid when the dendritic tree receives an ongoing barrage of input. To do so, we construct a control NET where the leafs are completely independent by replacing Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (Equation 2), the sum of all voltage components associated with nodes that integrated more than one region) in the synaptic currents by their long-term average (here computed as a 200 ms low-pass filter of the root voltage under Poisson stimulation with a fixed rate). Consequently, in this independent NET (iNET), leaf voltage components (and thus the synaptic currents) only depend on the local inputs by construction. Convergence between the iNET and the normal NET then indicates independence between input regions. We illustrate the iNET in a toy model with two leafs and one root (Figure 4A). When a synaptic input arrives at synapse 1, an NMDA event will be generated in branch 1. The leaf voltage associated with this event is shown in the leftmost panels in Figure 4B. The effect of a strong input at synapse 2 on the dynamics in branch 1 depends on Math Eq {focus_keyword} Electrical Compartmentalization in Neurons; for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, the shape of the NET voltage trace at leaf 1 is drastically modified, whereas at Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, there is little change (Figure 4B, rightmost panels). In the iNET, the traces at leaf 1 with or without input to synapse 2 are identical by construction. Hence, when the dynamics of the NET converge to those of the iNET, the associated input regions constitute independent subunits. We stimulated both input sites with Poisson inputs, and at Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, the iNET voltage traces at soma and leafs deviate significantly from their NET counterparts (Figure 4C, top). The cooperative dynamics between input sites are thus nonnegligible. At Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, the iNET and NET dynamics agree very well (Figure 4C, bottom). Analyzing the correlations between both leaf voltage components confirms this finding; for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, the NET correlation approaches the iNET value of zero (Figure 4D). The RMSE between the somatic traces also vanishes for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. These results generalize to realistic neuron models and multiple input regions; analysis of leaf voltage correlations and somatic RMSEs for NETs obtained by distributing four input regions on the pyramidal cell morphology (Figure 2C) yields similar results (Figure 4E; x axis represents the average Math Eq {focus_keyword} Electrical Compartmentalization in Neurons between the input regions). Furthermore, these results are robust for different equilibrium potentials (Figure 4F) and different voltage dependencies of the NMDA current (Figure 4G). We thus conclude, as a rule of the thumb, that for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, pairs of dendritic regions can be considered independent.

Figure thumbnail gr4 {focus_keyword} Electrical Compartmentalization in Neurons

Figure 4The Impedance-Based Independence Index Math Eq {focus_keyword} Electrical Compartmentalization in Neurons Leads to a Systematic Characterization of Independent Subunits

Show full caption

(A) Toy model morphology and associated NET.

(B) Voltage associated with leaf node 1 for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (top) and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (bottom). iNET (red) and NET (blue) traces for a single input to synapse 1 (left) and for inputs to synapses 1 and 2 (right). Vertical black lines indicate input arrival.

(C) Same model, but now both synapses receive Poisson inputs. Somatic voltage (here equal to Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, left) and node voltage (right) are shown for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (top) and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (bottom).

(D) The correlation between leaf voltage components of NET and iNET (black), and their somatic RMSE (red), as a function of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Same model as in (A)–(C).

(E) Same as in (D), but with 4 inputs regions selected on the L5 pyramidal cell in each simulation. The x axis represents the average Math Eq {focus_keyword} Electrical Compartmentalization in Neurons between regions.

(F) Correlation between leaf voltage components of NET (decreasing line) and iNET (flat lines) for different equilibrium potentials. Same model as in (A)–(C).

(H) Cortical stellate cell.

(I) Hippocampal granule cell. For these cells, morphology is shown on the right and associated matrix of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons values on the left.

(J) Cortical L5 pyramidal cell. Morphology (left) with color-coded independent regions at Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and NET with the same color-coded independent subunits (right). Note that individual compartments are always connected; color matches between separated compartments are accidental.

We then analyzed the asymptotic equilibrium points of the NMDA dynamics. With these points, the dynamic aspects of NMDA spikes can be fully understood (

Major et al., 2008

  • Major G.
  • Polsky A.
  • Denk W.
  • Schiller J.
  • Tank D.W.

Spatiotemporally graded NMDA spike/plateau potentials in basal dendrites of neocortical pyramidal neurons.

), as they represent the voltage a dendritic branch is trying to reach given the input to NMDA synapses. Nevertheless, influence of neighboring inputs on these points has never been investigated. We developed a method to compute these points for any input configuration (see Methods S1) and studied at which Math Eq {focus_keyword} Electrical Compartmentalization in Neurons values a fully developed NMDA spike can form independently in a branch (Figure S2B). We also asked at which Math Eq {focus_keyword} Electrical Compartmentalization in Neurons values two NMDA spikes would sum together, as they would for independent subunits (Figure S2C). Finally, we investigated the effective shift in potentiation threshold as a function of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (Figures S2D and S2E). These analyses corroborated our conclusion that, for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons dendritic regions function as independent subunits.

 A Formal Definition of Dendritic Compartmentalization

Next, we asked how many independent regions could maximally coexist along a dendritic tree, as well as what their location would be. We constructed an algorithm that divides the dendritic tree into regions separated by a minimal Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (see STAR Methods). For the stellate and granule cells, we did not identify any compartments, as there were no Math Eq {focus_keyword} Electrical Compartmentalization in Neurons values greater than 10 in these cells (Figures 4H and 4I). In the pyramidal cell, the number of compartments for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons was far less than the number of dendritic terminals (Figure 4J). Even at lower Math Eq {focus_keyword} Electrical Compartmentalization in Neurons values, there are still much fewer independent subunits than dendritic terminals (Figure S3; see, for instance, compartment numbers for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons). Note in this context that compartment numbers are fairly robust with respect to the biophysical parameters (Figure S4A); increasing axial resistance and membrane conductance, as well as adding a spine correction factor, increase compartment numbers slightly, whereas increasing the dendritic radii results in a moderate decrease of compartment numbers. We conclude that because cooperativity between synapses decreases less than voltage attenuation as a function of electrical separation, the number of independent subunits that can coexist on a dendritic tree is lower than the number of dendritic branches. Our results thus contradict the notion that every branch constitutes an independent electrical subunit.

 Comparison of the Model to the Electrical Properties of Basal Dendrites

In order to validate if the predicted Math Eq {focus_keyword} Electrical Compartmentalization in Neurons values, and thus the degree of electrical compartmentalization, are physiologically plausible, we reanalyzed dual dendritic patch-clamp recordings from the basal dendrites of neocortical L5 pyramidal neurons (

Nevian et al., 2007

  • Nevian T.
  • Larkum M.E.
  • Polsky A.
  • Schiller J.

Properties of basal dendrites of layer 5 pyramidal neurons: a direct patch-clamp recording study.

). We computed Math Eq {focus_keyword} Electrical Compartmentalization in Neurons between dendritic branches with similar electrical properties emanating from the soma (Figure 5A). We corrected the recorded traces for experimental inaccuracies in access resistance compensation (Figure S5) and extracted steady-state responses during hyperpolarizing current pulses to infer input and transfer impedances (Figures 5B and 5C). Error bars on the input impedance values were determined from the accuracy of the access resistance fit to estimate the experimental variability (Figure S5C). The uncompensated recordings gave quantitatively similar results. We estimated Math Eq {focus_keyword} Electrical Compartmentalization in Neurons according to Equation 4 and found that the values agreed well with predictions from the L5 pyramidal cell model (Figure 5D). We consequently conclude that our predictions on compartment numbers (as long as the threshold of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons predicted by the models is correct) are compatible with experimental data.

Figure thumbnail gr5 {focus_keyword} Electrical Compartmentalization in Neurons

Figure 5Estimating Math Eq {focus_keyword} Electrical Compartmentalization in Neurons from Dendritic Patch-Clamp Data

Show full caption

(A) Schematic depiction of the analysis. Input and transfer (both dendrite to soma and soma to dendrite) impedance values were estimated from simultaneous dendritic and somatic voltage recordings. Input current steps were either injected at the dendritic or the somatic electrode. The extracted impedances were then used to compute what Math Eq {focus_keyword} Electrical Compartmentalization in Neurons would be between the dendritic electrode site and a site at equal distance from the soma on a hypothetical dendritic branch with similar electrical properties.

(B) Example voltage response traces. Voltage difference was computed as Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, where Math Eq {focus_keyword} Electrical Compartmentalization in Neurons is the average access-resistance-corrected voltage between 200 and 350 ms after stimulus onset and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons is the average voltage in the 50 ms before onset.

(C) Impedance values were extracted from the slope of the regression line between input current (here delivered at the dendritic electrode) and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (data extracted from traces shown in B). Error bars were determined from the access resistance fit (Figure S5).

(D) Math Eq {focus_keyword} Electrical Compartmentalization in Neurons as a function of the distance of the dendritic electrode to the soma (red, experimental data; gray, L5 pyramidal cell).

 Dendritic Compartmentalization Can Be Modified Dynamically by Inputs

As neurons perform different input-output transformations at different moments in time, such as during up- and down-states (

Wilson and Kawaguchi, 1996

  • Wilson C.J.
  • Kawaguchi Y.

The origins of two-state spontaneous membrane potential fluctuations of neostriatal spiny neurons.

), we hypothesized that the number of compartments in dendrites can be modified dynamically by input patterns—an idea corroborated by recent experimental results (

Cichon and Gan, 2015

  • Cichon J.
  • Gan W.-B.

Branch-specific dendritic Ca(2+) spikes cause persistent synaptic plasticity.

,

Poleg-Polsky et al., 2018

  • Poleg-Polsky A.
  • Ding H.
  • Diamond J.S.

Functional compartmentalization within starburst amacrine cell dendrites in the retina.

). In principle, shunting inputs could mediate such a change. Indeed, a voltage component with impedance kernel Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, a shunting conductance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, and excitatory conductances Math Eq {focus_keyword} Electrical Compartmentalization in Neurons:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 5)

can be reinterpreted in the following way (see Methods S1):

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 6)

so that the shunt effectively reduces impedance (note that this relation is approximate, since, by rewriting Equation 5 in this way, we took Math Eq {focus_keyword} Electrical Compartmentalization in Neurons in the shunt term out of the convolution). When shunts are placed on the dendrite in such a way that they primarily affect shared nodes, independence between branches would increase.

In the high-conductance state (

Destexhe et al., 2003

  • Destexhe A.
  • Rudolph M.
  • Paré D.

The high-conductance state of neocortical neurons in vivo.

), a state occurring in vivo when many synapses are randomly activated, the conductance of the membrane is increased over the whole dendritic tree, implementing a global shunt. A priori, two things could happen: either the root impedance of the associated NET could decrease more than the leaf impedances, in which case the number of subunits would increase, or the root impedance could decrease less than the leaf impedances, in which case the number of subunits would decrease. Because the root node of the associated NET integrates much more inputs than more local nodes, the former tends to decrease more than the latter. We illustrate this effect in the stellate cell, where a NET was derived for three input locations (Figure 6A): once at rest (Figure 6B, black NET) and once in the high conductance state (Figure 6B, blue NET, computed by adding the time-average conductance of the background synapses to the membrane as static shunts). Because root impedance decreases more than leaf impedance, electrical separation becomes stronger between branches that only have the root in common, and compartments emerge in the stellate cell (Figure 6C). The high-conductance NET is accurate in reproducing the average voltages; traces in Figure 6D (full colored lines) agree very well with the average post-synaptic potentials computed during an ongoing barrage of balanced excitation and inhibition (dashed colored lines).

Figure thumbnail gr6 {focus_keyword} Electrical Compartmentalization in Neurons

Figure 6Dynamic Compartmentalization due to Spatiotemporal Input Patterns

Show full caption

(A) Stellate cell morphology with three input regions (numbers 1–3) and somatic readout.

(B) NET for the configuration in (A) at rest (black) and in the high-conductance state (blue).

(C) Compartmentalization for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons in the rest state (top) versus the high conductance state (bottom).

(D) At the three regions of interest, a strong excitatory synapse was inserted. The average post-synaptic potential was computed over 100 trials (dashed line) and coincides with the NET prediction (full line). Responses without background activity are plotted in black for reference.

(E) Apical tuft of the L5 pyramidal cell, where we studied the effect of inhibition on the compartmentalization.

(F) NET associated with the apical tuft, without (top) and with (bottom) inhibition (with an time-averaged conductance of 5 nS).

(G) Voltage traces at synapses 3 and 4 without (top) and with (bottom) shunting inhibition (black trace is the equivalent neuron simulation).

(H and I) Math Eq {focus_keyword} Electrical Compartmentalization in Neurons change when inhibition is turned on (H), and associated change in membrane correlation when synapses in both branches were stimulated with random Poisson trains (I).

Since both theory (

Gidon and Segev, 2012

  • Gidon A.
  • Segev I.

Principles governing the operation of synaptic inhibition in dendrites.

) and connectivity data (

Bloss et al., 2016

  • Bloss E.B.
  • Cembrowski M.S.
  • Karsh B.
  • Colonell J.
  • Fetter R.D.
  • Spruston N.

Structured dendritic inhibition supports branch-selective integration in CA1 pyramidal cells.

) suggest the importance of the precise location of synaptic inhibition, we investigate the influence of precisely located inhibitory inputs on compartmentalization. In the apical tree of the L5 pyramidal neuron (Figure 6E), we noticed that the sibling branches in a particular apical fork did not constitute separate compartments (locations 3 and 4 in Figure 6F). Upon inserting inhibitory synapses near the branch point between the two terminal segments, they separate into independent subunits (Figure 6F, bottom). The required inhibition (with a time-averaged conductance of 5 nS) could be provided by the somatostatin-positive interneuron pathways targeting the apical tuft (

Markram et al., 2015

  • Markram H.
  • Muller E.
  • Ramaswamy S.
  • Reimann M.W.
  • Abdellah M.
  • Sanchez C.A.
  • Ailamaki A.
  • Alonso-Nanclares L.
  • Antille N.
  • Arsever S.
  • et al.

Reconstruction and simulation of neocortical microcircuitry.

,

Muñoz et al., 2017

  • Muñoz W.
  • Tremblay R.
  • Levenstein D.
  • Rudy B.

Layer-specific modulation of neocortical dendritic inhibition during active wakefulness.

). This change in compartmentalization can be quantified by the change in Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (Figure 6H). Note that the effect is location specific; independence between locations 1 and 2 does not increase. We performed simulations where each synapse pair received Poisson inputs (Figure 6G; note that NET traces again agree with the neuron simulation) and computed the correlations between both pairs of synapses (Figure 6I). The decrease in correlation upon activation of the shunting inhibition is consistent with the reduction in Math Eq {focus_keyword} Electrical Compartmentalization in Neurons.

 Dynamic Compartmentalization Can Enable Branch-Specific Learning

Recent experiments have demonstrated that inhibitory interneurons are required for branch-specific plasticity (

Cichon and Gan, 2015

  • Cichon J.
  • Gan W.-B.

Branch-specific dendritic Ca(2+) spikes cause persistent synaptic plasticity.

). Can a transient recompartmentalization, mediated by inhibition, underlie this branch-specific learning? Before learning, post-synaptic targeting is thought to be unspecific (

Gerstner et al., 1996

  • Gerstner W.
  • Kempter R.
  • van Hemmen J.L.
  • Wagner H.

A neuronal learning rule for sub-millisecond temporal coding.

). Hence, inputs coding different stimulus features can arrive at the same branch, but with different strengths. We ask whether sibling branches in the apical tree of the L5 pyramidal neuron (Figure 7A) can learn to become selective only to the strongest initial feature (Figure 7A) using only NMDA spikes and no APs (

Hardie and Spruston, 2009

  • Hardie J.
  • Spruston N.

Synaptic depolarization is more effective than back-propagating action potentials during induction of associative long-term potentiation in hippocampal pyramidal neurons.

). If two branches receive different synaptic activation (quantified as the product between input impedance and synaptic conductance), then the voltage difference between these branches will be larger when they are separated by a higher Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (Figure 7B), and therefore the probability increases to robustly potentiate the preferred branch while the nonpreferred branch is depressed. For the selected branches, activation of inhibitory synapses near the bifurcation point increased Math Eq {focus_keyword} Electrical Compartmentalization in Neurons from 3.0 to 8.5 (Figure 7C, black versus blue tree). The original Math Eq {focus_keyword} Electrical Compartmentalization in Neurons value was too low (Figures 7D and 7G), resulting in a positively correlated weight evolution in the preferred and nonpreferred branches (Figure 7F). Activation of inhibitory synapses near the bifurcation point increased Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, thus anticorrelating the weight evolution (Figure 7F) and enabling branch-specific learning (Figures 7E and 7G).

Figure thumbnail gr7 {focus_keyword} Electrical Compartmentalization in Neurons

Figure 7Enabling Branch-Specific Learning with Shunting Conductances

Show full caption

(A) Sketch of the situation. Before learning, two sibling branches are both targeted by synapses coding for either an up stimulus Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (orange) or a down stimulus Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (green). One branch Math Eq {focus_keyword} Electrical Compartmentalization in Neurons receives more synapses of the former stimulus, and the other branch Math Eq {focus_keyword} Electrical Compartmentalization in Neurons receives more of the latter stimulus. Branches should learn to be sensitive only to the stimulus that was initially the most prevalent (the “preferred” stimulus).

(B) Difference in voltage across sibling branches Math Eq {focus_keyword} Electrical Compartmentalization in Neurons increases as a function of the difference in activation Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. For higher Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, this increase is steeper. Consequently, a given threshold Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (black dashed line) is reached for lower Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (colored vertical lines).

(C) Schematic of the NET without (black) and with (blue) shunt.

(D and E) The learning task without (D; effective Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) or with (E; effective Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) shunting inhibition. For each epoch, both stimuli are presented for 100 ms, with 150-ms intervals in between them, for a total of 20 epochs. The initial and final epochs are shown.

(F) Correlation between the average weights of the synapses in the preferred and nonpreferred branches during stimulus presentation, averaged over all epochs and 20 repetitions of the learning task.

(G) Bar plot of the weight difference after the final epoch for all repetitions of the learning task. The bar lengths denote the medians of the weight difference distributions and the error bars the 25–75 percentiles.

Discussion

In this work, we formalized electrical dendritic compartmentalization. We have shown that dendritic regions are independent if the fluctuations in global NET voltage components are small compared to the fluctuations in local NET voltage components. Furthermore, the relative sizes of these fluctuations are tightly related to the sizes of the associated NET impedances. We have thus proposed Math Eq {focus_keyword} Electrical Compartmentalization in Neurons as a ratio of these impedances and have shown that this number is a good and measurable predictor of independence. We found that for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, pairs of dendritic sites function as independent subunits. Furthermore, we have used the NET to design an algorithm that, given a threshold Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, yields the maximal number of regions that can coexist on the dendritic tree separated by at least this threshold Math Eq {focus_keyword} Electrical Compartmentalization in Neurons.

We have then performed this analysis on a number of cell classes (Figure S3) and found, in line with another recent study (

Ujfalussy et al., 2018

  • Ujfalussy B.B.
  • Makara J.K.
  • Lengyel M.
  • Branco T.

Global and multiplexed dendritic computations under in vivo-like conditions.

), that many branches are not separated by Math Eq {focus_keyword} Electrical Compartmentalization in Neurons values required for independence. Impedance drops across dendritic bifurcations lead to large voltage attenuation (Figure 3), and studying simple input scenarios may lead to the idea that sibling branches automatically constitute independent subunits. However, in realistic scenarios where both branches receive an ongoing barrage of inputs, small transfer impedance values still lead to a large degree of cooperativity between synapses (Figure 4). Only for large electrical separations Math Eq {focus_keyword} Electrical Compartmentalization in Neurons are branches able to function independently. Although it is striking that a simple ratio of impedances can predict whether responses will be independent, earlier work has already hinted at the importance of impedances to explain dendritic response properties (

Vetter et al., 2001

  • Vetter P.
  • Roth A.
  • Häusser M.

Propagation of action potentials in dendrites depends on dendritic morphology.

).

Experimental data on the electrical coupling in fine dendritic branches, as required to support our theoretical findings, are rare. Nevertheless, we wondered whether our predictions could be validated. We found that existing recordings from fine basal dendrites of L5 pyramidal neurons in the neocortex allowed for the estimation of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons between different main branches emanating from the soma (Figure 5). The values we extracted agreed well with predictions by our model, thus validating that the latter lie within a biologically plausible range and confirming our predictions on compartmentalization. Ideally, to directly validate our prediction of the relation between compartmentalization and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, triple dendritic recordings of parent and higher-order daughter branches would have to be combined with the focal synaptic uncaging of glutamate to elicit local NMDA spikes or other regenerative events. A related approach would be to perform dual dendritic recordings in vivo to determine dendritic independence in the most natural state, yet these experiments are challenging and have not been performed so far.

We focused on electrical compartmentalization of dendrites and not on chemical compartmentalization. The latter might be more localized than the former, a classical example being the localization of Ca2+ signals within the spine head but not the neighboring dendrite (

Nevian and Sakmann, 2004

  • Nevian T.
  • Sakmann B.

Single spine Ca2+ signals evoked by coincident EPSPs and backpropagating action potentials in spiny stellate cells of layer 4 in the juvenile rat somatosensory barrel cortex.

,

Yuste and Denk, 1995

  • Yuste R.
  • Denk W.

Dendritic spines as basic functional units of neuronal integration.

). Stronger chemical compartmentalization may lead to localized plasticity (

Govindarajan et al., 2011

  • Govindarajan A.
  • Israely I.
  • Huang S.-Y.
  • Tonegawa S.

The dendritic branch is the preferred integrative unit for protein synthesis-dependent LTP.

,

Losonczy et al., 2008

  • Losonczy A.
  • Makara J.K.
  • Magee J.C.

Compartmentalized dendritic plasticity and input feature storage in neurons.

,

Weber et al., 2016

  • Weber J.P.
  • Andrásfalvy B.K.
  • Polito M.
  • Magó Á.
  • Ujfalussy B.B.
  • Makara J.K.

Location-dependent synaptic plasticity rules by dendritic spine cooperativity.

) and small spine cluster sizes (

Frank et al., 2018

  • Frank A.C.
  • Huang S.
  • Zhou M.
  • Gdalyahu A.
  • Kastellakis G.
  • Silva T.K.
  • Lu E.
  • Wen X.
  • Poirazi P.
  • Trachtenberg J.T.
  • Silva A.J.

Hotspots of dendritic spine turnover facilitate clustered spine addition and learning and memory.

,

Fu et al., 2012

  • Fu M.
  • Yu X.
  • Lu J.
  • Zuo Y.

Repetitive motor learning induces coordinated formation of clustered dendritic spines in vivo.

,

Gökçe et al., 2016

  • Gökçe O.
  • Bonhoeffer T.
  • Scheuss V.

Clusters of synaptic inputs on dendrites of layer 5 pyramidal cells in mouse visual cortex.

) that could not be explained by electrical compartmentalization alone.

The heterogeneity of brain states (

Wilson and Kawaguchi, 1996

  • Wilson C.J.
  • Kawaguchi Y.

The origins of two-state spontaneous membrane potential fluctuations of neostriatal spiny neurons.

) and the observation that the electrical length of nerve fibers changes with the amount of background conductance (

) led us to explore the possibility that the NET, and hence independence, could be modified dynamically by spatiotemporal input patterns. Up states (

Destexhe et al., 2007

  • Destexhe A.
  • Hughes S.W.
  • Rudolph M.
  • Crunelli V.

Are corticothalamic ‘up’ states fragments of wakefulness?.

) can increase membrane conductance across the neuron, as shown by simultaneous dendritic and somatic recordings in vivo (

Waters and Helmchen, 2004

  • Waters J.
  • Helmchen F.

Boosting of action potential backpropagation by neocortical network activity in vivo.

). In such cases, the impedance associated with the root node of the NET decreases more compared to impedances associated with leaf nodes. This suggests that independence across branches increases during up states (Figures 6A–6D). Nevertheless, previous work on the dynamics in dendrites during such states (

Farinella et al., 2014

  • Farinella M.
  • Ruedt D.T.
  • Gleeson P.
  • Lanore F.
  • Silver R.A.

Glutamate-bound NMDARs arising from in vivo-like network activity extend spatio-temporal integration in a L5 cortical pyramidal cell model.

,

) suggests that it in certain cases, it becomes easier to elicit dendritic regenerative events and also that the set of synapses required to trigger such an event can be more spatially distributed. This apparent paradox is resolved by realizing that branches can only act independently if the fluctuations in the global voltage component are small compared to fluctuations in the local voltage components (Equation 2). A synaptic bombardment during the high-conductance state has two effects: the increased synaptic conductance dampens fluctuations in the global voltage components, decoupling dendritic regions, and random noise increases these fluctuations, coupling dendritic regions. The final picture thus depends on the balance between these two effects, which may change according to brain state, region, and even across different regions on the same cell.

The interaction of inhibition with NMDA spikes has been studied extensively. It is now known that inhibition is well suited to veto NMDA-spike generation (

Gidon and Segev, 2012

  • Gidon A.
  • Segev I.

Principles governing the operation of synaptic inhibition in dendrites.

,

Rhodes, 2006

  • Rhodes P.

The properties and implications of NMDA spikes in neocortical pyramidal cells.

), and allows for rapid switching between the on and off states of the NMDA plateau (

Doron et al., 2017

  • Doron M.
  • Chindemi G.
  • Muller E.
  • Markram H.
  • Segev I.

Timed synaptic inhibition shapes NMDA spikes, influencing local dendritic processing and global I/O properties of cortical neurons.

). Our work adds yet a third interaction to the repertoire, where inhibition (located near dendritic bifurcations) can decouple NMDA-spike generation in neighboring branches in a highly localized fashion, resulting in a precise tuning of compartmentalization (Figures 6E–6H).

How many branches on the dendritic tree lie within in the useful range for dynamic recompartmentalization? In pyramidal cells, compartments numbers doubled for values of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons between Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, and in smaller cells, these numbers increase 5-fold (Figure S3). This suggests that global conductance input increases the number of compartments 2- to 5-fold, depending on the cell’s properties. Inspecting pairwise independence, we determine that in pyramidal cells, 5% to 10% of terminal pairs are separated by Math Eq {focus_keyword} Electrical Compartmentalization in Neurons values between 3 and 10 (mainly terminals on the same main branches). In smaller cells, up to 60% of pairs fall within these values. On average, these pairs are made independent by shunting conductances of 5 to 15 nS (Figure S3).

To explore the functional consequences of dynamic compartmentalization, we equipped synapses on dendritic sibling branches with a voltage-based plasticity rule (

Clopath et al., 2010

  • Clopath C.
  • Büsing L.
  • Vasilaki E.
  • Gerstner W.

Connectivity reflects coding: a model of voltage-based STDP with homeostasis.

) and explored whether these synapses could learn independently. Consistent with our prior observations, we found that a value of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons was too small for these branches to learn independently. Nevertheless, shunting inhibition that increased Math Eq {focus_keyword} Electrical Compartmentalization in Neurons to Math Eq {focus_keyword} Electrical Compartmentalization in Neurons allowed the synaptic weights in both branches to evolve in an independent fashion. Interestingly, recent data suggest such phenomena may occur in vivo (

Cichon and Gan, 2015

  • Cichon J.
  • Gan W.-B.

Branch-specific dendritic Ca(2+) spikes cause persistent synaptic plasticity.

). Thus, the computation a neuron is engaged in may vary across brain states; when background conductance is high, neurons may prioritize in local dendritic learning, whereas otherwise, they may favor associative output generation.

The NET framework relies on two approximations: that impedance kernels with similar magnitudes share similar time-scales, so that they can be averaged, and that centripetal voltage attenuation is small compared to centrifugal attenuation (

Nevian et al., 2007

  • Nevian T.
  • Larkum M.E.
  • Polsky A.
  • Schiller J.

Properties of basal dendrites of layer 5 pyramidal neurons: a direct patch-clamp recording study.

). We found these approximations to be true in all cortical neuron types we modeled. Furthermore, the NET framework is not restricted to the currents we modeled; Ca-spike generating channels can be included as well (

Larkum et al., 1999

  • Larkum M.E.
  • Zhu J.J.
  • Sakmann B.

A new cellular mechanism for coupling inputs arriving at different cortical layers.

). NETs express the interaction of a (sub)set of synapses in a background determined by the electrical properties of a morphology. Hence, there is freedom in choosing which synapses to model explicitly and which synapses to treat as background by including their average effect in the NET. This choice depends on the problem at hand. For instance, we have shown here that modeling shunting inhibition implicitly yields insight in the change in interaction between NMDA synapses.

Finally, we have devised an efficient inversion algorithm (see Methods S1), so that performant NET simulations can be designed. While for full neuron models, the standard simulation tools remain the preferable option (

), the NET framework allows for the definition of abstract dendrite models so that hallmark dendritic computations can be implemented in a minimal fashion. NETs may thus prove useful to scientists exploring the effects of dendrites at the network level.

Across the brain, neurons take on a wide variety of morphologies. We have shown here how these dendritic trees compartmentalize at rest and during dynamic input regimes. The behavioral relevance of up states (

Destexhe et al., 2007

  • Destexhe A.
  • Hughes S.W.
  • Rudolph M.
  • Crunelli V.

Are corticothalamic ‘up’ states fragments of wakefulness?.

), the specificity of inhibitory targeting (

Bloss et al., 2016

  • Bloss E.B.
  • Cembrowski M.S.
  • Karsh B.
  • Colonell J.
  • Fetter R.D.
  • Spruston N.

Structured dendritic inhibition supports branch-selective integration in CA1 pyramidal cells.

), and the importance of interneuron activity for branch-specific learning (

Cichon and Gan, 2015

  • Cichon J.
  • Gan W.-B.

Branch-specific dendritic Ca(2+) spikes cause persistent synaptic plasticity.

) suggest that dynamic compartmentalization is ubiquitous in normal brain function, with far-reaching consequences for memory formation (

Kastellakis et al., 2016

  • Kastellakis G.
  • Silva A.J.
  • Poirazi P.

Linking memories across time via neuronal and dendritic overlaps in model neurons with active dendrites.

) and capacity (

Poirazi and Mel, 2001

  • Poirazi P.
  • Mel B.W.

Impact of active dendrites and structural plasticity on the memory capacity of neural tissue.

). Taken together, the NET can be seen as a computational description of the morphological neuron, complementary to the well-known biophysical description, and the algorithm to derive it as a translation from biophysics to computation.

STAR★Methods

 Key Resources Table

 Contact for Reagent and Resource Sharing

Further information and requests for resources and reagent should be directed to and will be fulfilled by the Lead Contact, Benjamin Torben-Nielsen (btorbennielsen@gmail.com).

 Method Details

 Biophysical modeling

 Morphologies

Three exemplar morphologies were used for the analysis: a cortical stellate cell (

Wang et al., 2002

  • Wang Y.
  • Gupta A.
  • Toledo-Rodriguez M.
  • Wu C.Z.
  • Markram H.

Anatomical, physiological, molecular and circuit properties of nest basket cells in the developing somatosensory cortex.

) (Figure 2A), a hippocampal granule cell (

Carim-Todd et al., 2009

  • Carim-Todd L.
  • Bath K.G.
  • Fulgenzi G.
  • Yanpallewar S.
  • Jing D.
  • Barrick C.A.
  • Becker J.
  • Buckley H.
  • Dorsey S.G.
  • Lee F.S.
  • Tessarollo L.

Endogenous truncated TrkB.T1 receptor regulates neuronal complexity and TrkB kinase receptor function in vivo.

) (Figure 2B) and a cortical pyramidal cell (

Hay et al., 2011

  • Hay E.
  • Hill S.
  • Schürmann F.
  • Markram H.
  • Segev I.

Models of neocortical layer 5b pyramidal cells capturing a wide range of dendritic and perisomatic active properties.

) (Figure 2C). These morphologies were retrieved from the NeuroMorpho.org repository (

Ascoli, 2006

  • Ascoli G.A.

Mobilizing the base of neuroscience data: the case of neuronal morphologies.

), except the pyramidal cell, which was retrieved from the ModelDB repository (

Hines et al., 2004

  • Hines M.L.
  • Morse T.
  • Migliore M.
  • Carnevale N.T.
  • Shepherd G.M.

ModelDB: a database to support computational neuroscience.

). Cell morphologies used in our wider cortical analysis were retrieved from the Blue Brain Project database (

Markram et al., 2015

  • Markram H.
  • Muller E.
  • Ramaswamy S.
  • Reimann M.W.
  • Abdellah M.
  • Sanchez C.A.
  • Ailamaki A.
  • Alonso-Nanclares L.
  • Antille N.
  • Arsever S.
  • et al.

Reconstruction and simulation of neocortical microcircuitry.

).

 Physiological parameters

Physiological parameters for the morphologies were set according to Major et al. (

Major et al., 2008

  • Major G.
  • Polsky A.
  • Denk W.
  • Schiller J.
  • Tank D.W.

Spatiotemporally graded NMDA spike/plateau potentials in basal dendrites of neocortical pyramidal neurons.

): the equilibrium potential was Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, the membrane conductance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, the capacitance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and the intracellular resistance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons.

To generate somatic APs, we used the fast inactivating Math Eq {focus_keyword} Electrical Compartmentalization in Neurons current Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and the fast, non-inactivating Math Eq {focus_keyword} Electrical Compartmentalization in Neurons current Math Eq {focus_keyword} Electrical Compartmentalization in Neurons previously employed in cortical models (

Hay et al., 2011

  • Hay E.
  • Hill S.
  • Schürmann F.
  • Markram H.
  • Segev I.

Models of neocortical layer 5b pyramidal cells capturing a wide range of dendritic and perisomatic active properties.

). Channel densities were: Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. The leak current was then fitted to yield a membrane timescale of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and an equilibrium potential of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons.

AMPA and GABA synaptic input currents were modeled as the product of a conductance profile, given by a double exponential shape (

Rotter and Diesmann, 1999

  • Rotter S.
  • Diesmann M.

Exact digital simulation of time-invariant linear systems with applications to neuronal modeling.

), with a driving force (

Jack et al., 1975

  • Jack J.J.
  • Noble D.
  • Tsien R.W.

Electric Current Flow in Excitable Cells.

):

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 7)

For AMPA synapses, we used rise resp. decay times Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, Math Eq {focus_keyword} Electrical Compartmentalization in Neurons for the conductance window and a reversal potential Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, while for GABA synapses we used Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. For N-methyl-D-aspartate (NMDA) channels (

Jahr and Stevens, 1990a

  • Jahr C.E.
  • Stevens C.F.

A quantitative description of NMDA receptor-channel kinetic behavior.

,

MacDonald and Wojtowicz, 1982

  • MacDonald J.F.
  • Wojtowicz J.M.

The effects of L-glutamate and its analogues upon the membrane conductance of central murine neurones in culture.

), the synaptic current had the form

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 8)

and the rise resp. decay time were Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, while Math Eq {focus_keyword} Electrical Compartmentalization in Neurons was the sigmoidal function employed by (

Behabadi and Mel, 2014

  • Behabadi B.F.
  • Mel B.W.

Mechanisms underlying subunit independence in pyramidal neuron dendrites.

) to model the channels’ magnesium block:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 9)

We also tested our results with the original gating function by

Jahr and Stevens (1990b)

  • Jahr C.E.
  • Stevens C.F.

Voltage dependence of NMDA-activated macroscopic conductances predicted by single-channel kinetics.

:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 10)

and the voltage dependence of the NMDA conductance in spiny stellate cells (

Lavzin et al., 2012

  • Lavzin M.
  • Rapoport S.
  • Polsky A.
  • Garion L.
  • Schiller J.

Nonlinear dendritic processing determines angular tuning of barrel cortex neurons in vivo.

):

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 11)

In the remainder of this work, we will refer to the voltage-dependent factors in the synaptic input current as the ‘synaptic voltage dependence’ (SVD), denoted by Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Hence, for AMPA or GABA synapses

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 12)

and for NMDA synapses

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 13)

Note that when we refer to the conductance of a simple synapse, we mean the maximum value Math Eq {focus_keyword} Electrical Compartmentalization in Neurons of its conductance window. For a synapse that has AMPA and NMDA components (to which we will simply refer as an NMDA synapse), the conductance is the maximal value of the AMPA conductance window, and the conductance of the NMDA component is determined by multiplying the AMPA conductance value with an NMDA ratio Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, that was set to be either 2 or 3.

 Plasticity

In our simulations with plasticity, we use a voltage dependent spike timing dependent plasticity rule (

Bono and Clopath, 2017

  • Bono J.
  • Clopath C.

Modeling somatic and dendritic spike mediated plasticity at the single neuron and network level.

,

Clopath et al., 2010

  • Clopath C.
  • Büsing L.
  • Vasilaki E.
  • Gerstner W.

Connectivity reflects coding: a model of voltage-based STDP with homeostasis.

) where the evolution of the weight Math Eq {focus_keyword} Electrical Compartmentalization in Neurons of a given synapse depends both on the post-synaptic voltage and the presynaptic AP inputs. This leads to a synaptic current of the following form:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 14)

In all our simulations the initial weight Math Eq {focus_keyword} Electrical Compartmentalization in Neurons was 1 and, during the simulation, the weight could fluctuate in the interval Math Eq {focus_keyword} Electrical Compartmentalization in Neurons.

 Compartmental models

To construct and simulate compartmental models of the cells, we used the neuron simulator (

). Compartment sizes were set to be smaller than or equal to the size given by the lambda rule (

).

 Green’s function and the separation of variables

To derive NETs, we rely on the Green’s function (GF) (

Koch, 1998

  • Koch C.

Biophysics of Computation: Information Processing in Single Neurons (Computational Neuroscience).

,

Wybo et al., 2013

  • Wybo W.A.M.
  • Stiefel K.M.
  • Torben-Nielsen B.

The Green’s function formalism as a bridge between single- and multi-compartmental modeling.

,

Wybo et al., 2015

  • Wybo W.A.M.
  • Boccalini D.
  • Torben-Nielsen B.
  • Gewaltig M.-O.

A sparse reformulation of the Green’s function formalism allows efficient simulations of morphological neuron models.

) Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. The GF is a function of three variables: two locations Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons along the dendritic arborization and a temporal variable Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. We compute the GF in an exponential basis:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 15)

by using the separation of variables (SOV) method (

Major and Evans, 1994

  • Major G.
  • Evans J.D.

Solutions for transients in arbitrarily branching cables: IV. Nonuniform electrical parameters.

,

Major et al., 1993

  • Major G.
  • Evans J.D.
  • Jack J.J.

Solutions for transients in arbitrarily branching cables: I. Voltage recording with a somatic shunt.

). Note that it is a property of the cable equation that the GF is symmetric in the spatial coordinates (

Koch, 1998

  • Koch C.

Biophysics of Computation: Information Processing in Single Neurons (Computational Neuroscience).

), so that Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Usually, a fixed set of discrete locations relevant for the problem at hand is chosen on the neuron. Hence, the GF only needs to be evaluated at these locations, and a discrete set of temporal kernels is obtained. A member of this set will be denoted as Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, to highlight the difference between the now discrete indices Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and the continuous variable Math Eq {focus_keyword} Electrical Compartmentalization in Neurons.

To compute the output voltage Math Eq {focus_keyword} Electrical Compartmentalization in Neurons at location Math Eq {focus_keyword} Electrical Compartmentalization in Neurons for a given input current Math Eq {focus_keyword} Electrical Compartmentalization in Neurons at location Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, one needs to compute the convolution of the GF evaluated at Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons with this input current (

Koch, 1998

  • Koch C.

Biophysics of Computation: Information Processing in Single Neurons (Computational Neuroscience).

):

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 16)

for which we will use the shorthand

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 17)

Since Math Eq {focus_keyword} Electrical Compartmentalization in Neurons converts current into voltage, we will refer to it as an ‘impedance kernel.’ The total surface under the impedance kernel is the steady state impedance:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 18)

In the rest of the text, it will be understood that Math Eq {focus_keyword} Electrical Compartmentalization in Neurons without temporal coordinate refers to the steady state impedance – which we will simply call ‘the impedance’ for brevity – while Math Eq {focus_keyword} Electrical Compartmentalization in Neurons is the temporal impedance kernel. To unclutter the notations, we will not make this distinction for other variables; the temporal dependence will be omitted by default. Following this convention, Equation (17) will be written as Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, where it is implied that both Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons are time dependent quantities since Math Eq {focus_keyword} Electrical Compartmentalization in Neurons is the temporal impedance kernel. Conversely, writing Math Eq {focus_keyword} Electrical Compartmentalization in Neurons means that Math Eq {focus_keyword} Electrical Compartmentalization in Neurons is the steady state impedance value, and thus Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons will be steady state values too. Note that currents in this text will be expressed in nano ampere (nA) and voltages in milli volt (mV). Consequently, impedances will be in mega ohm (MMath Eq {focus_keyword} Electrical Compartmentalization in Neurons).

 Synaptic activation

The eventual steady state voltage Math Eq {focus_keyword} Electrical Compartmentalization in Neurons obtained after activating a synaptic conductance at location Math Eq {focus_keyword} Electrical Compartmentalization in Neurons depends for a large part on the input impedance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Following (17), it can be obtained as a solution of the equation

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 19)

This solution Math Eq {focus_keyword} Electrical Compartmentalization in Neurons is thus a function of the product of impedance and conductance:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 20)

We refer to this product as the synaptic activation Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and note that it is a dimensionless quantity. Consequently, it is a convenient quantity that does not depend on local morphological constraints to determine whether an input will be strong enough to reach a certain voltage threshold, for instance to elicit an NMDA spike or to potentiate a synapse.

 Neural Evaluation tree

 Mathematical formulation

 Derivation

To derive the NET, we order all locations along the dendritic arborization in a depth-first manner (

) (Figures S1A–S1C), so that the impedance matrix (

Cuntz et al., 2010

  • Cuntz H.
  • Forstner F.
  • Borst A.
  • Häusser M.

One rule to grow them all: a general theory of neuronal branching and its practical application.

) has a highly organized structure (Figures 1C and 1D). Generally an even blue surface covers most of the matrix, representing the transfer impedances between the main dendritic branches. There are also smaller square regions of light blue or green closer to the diagonal, representing sibling branches that are electrically closer to each other than to different main branches. Finally, the small squares along the diagonal colored yellow and red are the thin dendritic tips with high input impedances. Consequently, dendritic tips that lie within the same light blue or green square are closer together electrotonically than tips within different squares, as the transfer impedances connecting them are much higher. A NET tree graph structure hence imposes itself naturally: the dendritic tips constitute the leafs of the tree, their parent node combines multiple adjacent tips and the root node in turn binds all these nodes together. To derive the NET tree graph, we define an impedance step Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and execute the following recursively:

Generally, one aims to study a subset of “regions of interest” on the dendritic tree. In such cases, the original NET can be pruned so that a strongly reduced NET is obtained. The pruning consist of two operations: First, nodes that do not integrate regions of interest are removed. Second, nodes that integrate the same subset of the regions of interest are combined into a single node, whose impedance kernel is the sum of the impedance kernels of the original nodes. This reduced NET is then the minimal structure that faithfully captures the interactions between the regions of interest. As an example, if we would reduce the set of inputs Math Eq {focus_keyword} Electrical Compartmentalization in Neurons in Figure 1A to Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, the pink and green nodes both integrate the identical subset Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Hence, in the reduced NET, these nodes can be combined into a single node by the aforementioned procedure.

The NET framework can thus be understood as a discretization in “impedance,” instead of in space (as is the case for classical biophysical models). The average error of the NET approximation depends on Math Eq {focus_keyword} Electrical Compartmentalization in Neurons – in analogy to the average error in classical biophysical models, which depends on Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Nevertheless, because of the inherent approximations present in the NET framework, the average error does not go to zero but reaches a minimum value for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons MMath Eq {focus_keyword} Electrical Compartmentalization in Neurons (Figure S4B). Math Eq {focus_keyword} Electrical Compartmentalization in Neurons MMath Eq {focus_keyword} Electrical Compartmentalization in Neurons is thus a good choice for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons.

In order to simulate in the NET framework, we have derived an efficient simulation algorithm for Equation (21) (see Methods S1 for details)

 Distal and proximal regions

The aforementioned algorithm successfully constructs NETs of dendritic arborization where the variations in transfer impedance between branches are small compared to their average values (Figures S4C–S4E). If these variations are large, as is the case in pyramidal cells (Figure S4E), interactions between proximal and distal dendritic domains are overestimated. In pyramidal cells, distal domains are only connected to the soma by one or a few large dendritic branches. Hence, there are many points of low soma to dendrite impedance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, many of high Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, and relatively few of intermediate Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. By consequence, the histogram of all dendrite to soma transfer impedances has two main modes, whose boundary (

Delon et al., 2007

  • Delon J.
  • Desolneux A.
  • Lisani J.L.
  • Petro A.B.

A nonparametric approach for histogram segmentation.

) indicates the domains. Any region with Math Eq {focus_keyword} Electrical Compartmentalization in Neurons above this boundary will belong to the proximal domain (node Math Eq {focus_keyword} Electrical Compartmentalization in Neurons in Figure S4E), whereas connected regions with Math Eq {focus_keyword} Electrical Compartmentalization in Neurons below this boundary will constitute distal domains (colored red in Figure S4E, node Math Eq {focus_keyword} Electrical Compartmentalization in Neurons). Algorithmically, we determine the kernel of the root node as the average of all transfer impedance kernels between proximal and distal branches. Then, we start the recursive procedure as before, but with the impedance matrices restricted to the different domains Math Eq {focus_keyword} Electrical Compartmentalization in Neurons.

 Predicting spikes: linear terms

The effective NET transfer impedance between dendrite and soma is either constant or can take on a proximal and a distal value. Spike prediction is refined further by defining linear terms that capture the precise transfer impedance between input regions and soma. These kernels only contribute to the voltage at the soma, and thus have no influence on the intra-dendritic synaptic interactions. Mathematically, their contribution to the somatic voltage can be written as:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 22)

 Independence and compartmentalization

 Independence

The leafs of the NET only receive inputs from a single region. Nevertheless, they are not per se independent from the other synapses, since the SVD in their input current still depends on all nodal voltages on the path to the root:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 23)

This current will become truly independent from all other synapses if

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 24)

with Math Eq {focus_keyword} Electrical Compartmentalization in Neurons a (large) number that has to be determined empirically and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons denoting the short term fluctuations of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons around a long term average Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Here, short term means the time-scale on which neurons convert electrical inputs to output. As NMDA synapses have a decay time constant of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons ms, this averaging time-scale should be at least somewhat larger than the NMDA time constant (we chose Math Eq {focus_keyword} Electrical Compartmentalization in Neurons ms). Then (23) can be approximated as follows:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 25)

The long-term average in this equation is only influenced very little by the instantaneous values of the synaptic conductances and can hence be seen as a constant, i.e., a fixed parameter in the equation:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 26)

Consequently, the solution for Math Eq {focus_keyword} Electrical Compartmentalization in Neurons will not depend on the instantaneous values of the synaptic conductances at other locations.

 Estimating independence

Whether condition (24) holds depends on the structure of the NET as well as the relative size of the synaptic inputs. We assume that synaptic conductances in physiological regimes are of similar magnitude. In this case, condition (24) becomes a condition on the impedances:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 27)

where Math Eq {focus_keyword} Electrical Compartmentalization in Neurons denotes the number of regions node Math Eq {focus_keyword} Electrical Compartmentalization in Neurons integrates. When we are interested in determining whether a pair of regions Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (integrated by the leafs Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) can act independently, we can consider a reduced tree with two leafs, obtained by pruning all nodes associated with other regions. The new tree then has leaf impedances Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (i.e., a sum over impedances of nodes that integrate one region but not the other) and a root impedance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (i.e., a sum over impedances of nodes that integrate both regions). Then, regions Math Eq {focus_keyword} Electrical Compartmentalization in Neurons are independent if:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 28)

For mutual independence between Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, this equation has to hold for both Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. To summarize these two conditions in a single expression, we defined the ‘ impedance-based independence index’ Math Eq {focus_keyword} Electrical Compartmentalization in Neurons:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 29)

Then, if (28) holds for both regions, the following condition also holds:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 30)

Note that this is a necessary, but not a sufficient condition for mutual independence. However, as shown throughout the main manuscript, when asymmetry is not too high Math Eq {focus_keyword} Electrical Compartmentalization in Neurons is, despite its simplicity, a surprisingly accurate measure.

 Compartmentalization

Previously we discussed the conditions under which a single input site can be considered independent from the rest of the input regions. Nevertheless, when inputs are distributed in an almost continuous fashion along the dendritic arborization, such sites may not exist. It can be expected, however, that the structure of the dendritic tree favors a grouping of inputs, such that inputs belonging to different groups are all mutually independent but inputs belonging to the same group are not. A grouping of this type for homogeneously distributed inputs along the dendritic arborization, and where inputs belonging to different groups have an Math Eq {focus_keyword} Electrical Compartmentalization in Neurons above a certain threshold, will be called a compartmentalization of that dendritic tree for that given Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Note that in such a compartmentalization, not all input sites can belong to a group, as there will have to be at least some space between compartments.

How can such a compartmentalization be found? First, we remark that there is no unique answer to this question. Consider a forked dendritic tip. It may happen that inputs within each sister branch are independent from the rest of the dendritic tree, but the branches are not independent from each other. Furthermore, because of a steep impedance gradient within the branch, inputs at the bifurcation point may not be independent from the rest of the tree. Because of the first constraint, both tips can not form separate compartments, whereas because of the second constraint, they can not be grouped into a single compartment either. Hence only one branch can be chosen, and either choice forms a valid compartmentalization.

We implemented an algorithm that proposes, using the NET and given an Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, a compartmentalization that maximizes the number of compartments. Intuitively, the algorithm works by removing one leaf branch of a pair when the pair does not form separate compartments. By continuing this until every pair of leaf branches forms separate compartments, a maximal compartmentalization has been found. We note that if a node Math Eq {focus_keyword} Electrical Compartmentalization in Neurons in the NET tree forms a valid compartment, all nodes in the subtree of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons are part of the same compartment, since their Math Eq {focus_keyword} Electrical Compartmentalization in Neurons to other compartments will be higher than the Math Eq {focus_keyword} Electrical Compartmentalization in Neurons of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Hence, our algorithm will simply return a set of nodes, where it is understood that a compartment associated with a node from this set is its whole subtree. Our algorithm proceeds in three steps:

  • 1.

    We determine a ‘tentative’ compartmentalization. For each node N in the NET tree, we examine the bifurcation nodes Math Eq {focus_keyword} Electrical Compartmentalization in Neurons on the path Math Eq {focus_keyword} Electrical Compartmentalization in Neurons from N to the root. We check whether the following condition holds

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 31)

  • 2.

    with Math Eq {focus_keyword} Electrical Compartmentalization in Neurons the path from Math Eq {focus_keyword} Electrical Compartmentalization in Neurons to the root. If this condition is true for two nodes N and M that have B on their respective paths to the root, and where Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, these nodes will be separated by at least the required Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Hence, we say that Math Eq {focus_keyword} Electrical Compartmentalization in Neurons is a tentative compartment with respect to Math Eq {focus_keyword} Electrical Compartmentalization in Neurons.

  • 3.

    In a second step, we remove all leafs from the tree that could not possibly be separate compartments. To do so, we look at the highest order bifurcation Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and its child leafs. Then, if at least two child leafs are tentative compartments with respect to Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, the other leafs are removed. Otherwise, all child leafs but the one with largest impedance are removed. Note that in the latter case, Math Eq {focus_keyword} Electrical Compartmentalization in Neurons is not a bifurcation anymore and consequently will not induce tentative compartments. We continue to cycle through the bifurcation nodes of highest order until no more nodes can be removed.

  • 4.

    In a final step we assign the compartments. As we are now sure that every leaf is part of a separate compartment, we start at the leaf, find the nearest bifurcation node in the NET tree, and then recursively find the lowest order node that is still a tentative compartment of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. This node will be a compartment node in the final compartmentalization.

 Extracting IZ from patch-clamp data

 Extrapolating from two-electrode recordings

Although there was only data available from two-electrode patch clamp setups with one dendritic and one somatic electrode, it is still possible to extrapolate what the value of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons would be between two dendritic locations, if the second location Math Eq {focus_keyword} Electrical Compartmentalization in Neurons would be on a branch with similar electrical properties and at a similar distance from the soma as the first location Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. From the dendritic current injection, we can calculate the input impedance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and the dendrite-to-soma transfer impedance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons by fitting a regression line to the IV-curve. Conversely, from the somatic current injection, we can calculate the somatic input impedance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons as well as the soma-to-dendrite transfer impedance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons in a similar manner. We then assume that Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. Math Eq {focus_keyword} Electrical Compartmentalization in Neurons between these two sites is given by:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 32)

with Math Eq {focus_keyword} Electrical Compartmentalization in Neurons, Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons the transfer impedance between both dendritic sites. The latter impedance can be estimated from the transitivity property (

Koch, 1998

  • Koch C.

Biophysics of Computation: Information Processing in Single Neurons (Computational Neuroscience).

):

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 33)

 Estimating the dendritic access resistance

Full compensation of whole-cell current clamp recordings are difficult due to the positive feedback of the electronic compensation circuitry that can lead to ringing and disruption of the giga seal (

Brette and Destexhe, 2012

  • Brette R.
  • Destexhe A.

Intracellular recording. In Handbook of Neural Activity Measurement, R. Brette.

). This is particularly the case when high resistance pipettes have to be used as in the case of recordings from the basal dendrites of pyramidal neurons. The compensation of capacitance and access resistance during the experiment was performed following standard procedures aiming optimal compensation, while avoiding overcompensation and ringing. Additional offline compensation was achieved by fitting the initial change in membrane voltage at the dendritic and somatic recording site with a double exponential function function (

Anderson et al., 2000

  • Anderson J.S.
  • Carandini M.
  • Ferster D.

Orientation tuning of input conductance, excitation, and inhibition in cat primary visual cortex.

,

De Sa and MacKay, 2001

  • De Sa V.R.
  • MacKay D.J.C.

Model fitting as an aid to bridge balancing in neuronal recording.

):

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 34)

to the voltage transient after stimulus onset. To do so, we minimized a weighted sum of squares error

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 35)

where the weights were given by Math Eq {focus_keyword} Electrical Compartmentalization in Neurons (with Math Eq {focus_keyword} Electrical Compartmentalization in Neurons ms), a function chosen to prioritize accuracy at small Math Eq {focus_keyword} Electrical Compartmentalization in Neurons values (Figures S5A and S5B). The resulting fit Math Eq {focus_keyword} Electrical Compartmentalization in Neurons contained one fast and one slow time-scale (let us index the fast time-scale with 1), and we equated the superfluous voltage drop due to the access resistance to the prefactor of the fastest exponential:

Math Eq {focus_keyword} Electrical Compartmentalization in Neurons(Equation 36)

It can be seen that this analysis yielded a co-linear set of values for the different injected current amplitudes in each experiment (Figure 5C). An estimate of the access resistance was then obtained from the slope of the regression line between Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and the amplitude of the injected current step.

To estimate the variability of our approximations of the access resistance, we also performed the same procedure at the soma, where the access resistance for each experiment could be determined from the standard bridge-balance procedure. We then compared fitted resistances with the values yielded by bridge balance (Figure 5D) and found good agreement within the limits of experimental variability. The distribution of these values then provided an estimate of the variability in our fit (Figure S5D), which allowed us to compute the uncertainty of our access resistance fit and yielded the error flags (Figure S5E).

 Simulation-specific parameters

 Parameters Figure 1.

We evaluated the impedance matrix in panel B at 10 Math Eq {focus_keyword} Electrical Compartmentalization in Neuronsm intervals. In the simulation depicted in panels J and K, the synapses contained only an AMPA component with Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and no active channels were inserted in the soma.

 Parameters Figure 2.

100 NMDA (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) and 100 GABA (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) synapses were inserted on the morphology and activated with Poisson spike trains of 1 Hz.

 Parameters Figure 3.

The synapse at region 1 was an NMDA synapse with Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. The synapse received 5 input spikes in a 2.5 ms interval in order to trigger an NMDA spike

 Parameters Figure 4.

For the simulations in panels C-D, NMDA synapses (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) were used. Their Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and incoming Poisson rate were optimized to utilize the full range of the NMDA non-linearity. For the simulations in panel E, NMDA (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) and GABA (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) synapses were used and their rates were also optimized to utilize the full range of the NMDA non-linearity.

 Parameters Figure 6.

For the simulations in panel C, the main synapses contained only an AMPA component with Math Eq {focus_keyword} Electrical Compartmentalization in Neurons. To simulate the high-conductance state, 200 AMPA and 200 GABA synapses (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) were distributed evenly across the neuron. Each AMPA synapse was stimulated with a Poisson spike train of 5 Hz. The rate of stimulation for the GABA synapses was tuned to achieve a balanced input. To recompute the tree structures for panels B and D, the time-averaged conductances of all background synapses were inserted in the morphology as static shunts.

The inhibitory synapse in panels E-H had Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and was activated at a steady rate of 200 Hz, so that it’s total time-averaged conductance was around 5 nS. For the simulations in panel H, we inserted NMDA synapses (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) in both branches and stimulated them with a rate of 200 Hz. Note that these inputs could come from multiple presynaptic cells. Due to the linearity of the conductance dynamics however all spikes can be taken to add to the same conductance and can hence be modeled as a single synapse. The maximal conductance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons of the NMDA synapse was optimized to obtain an average depolarization of Math Eq {focus_keyword} Electrical Compartmentalization in Neurons in each branch, a target value which yields parameters that allow exploitation of the full range of the NMDA non-linearity.

 Parameters Figure 7.

In both simulations with and without shunting inhibition, noise was implemented at all three locations using AMPA (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) and GABA (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) synapses. Both were stimulated with Poisson spike trains of resp. 33 Hz and 83.1 Hz (tuned to achieve balance). The shunting inhibition in the parent branch was implemented by a GABA synapse (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) receiving a Poisson train with a rate of 277 Hz (tuned to reach a time-averaged conductance of 12 nS) during the 100 ms learning intervals. Note that this single conductance could again represent multiple synapses.

Stimulus-specific innervation patterns were: Math Eq {focus_keyword} Electrical Compartmentalization in Neurons to Math Eq {focus_keyword} Electrical Compartmentalization in Neurons: 5 synapses, Math Eq {focus_keyword} Electrical Compartmentalization in Neurons to Math Eq {focus_keyword} Electrical Compartmentalization in Neurons: 2 synapses, Math Eq {focus_keyword} Electrical Compartmentalization in Neurons to Math Eq {focus_keyword} Electrical Compartmentalization in Neurons: 2 synapses and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons to Math Eq {focus_keyword} Electrical Compartmentalization in Neurons: 5 synapses. These synapses were all NMDA synapses (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) that where activated in the learning intervals with Poisson trains at a rate of 33.3 Hz without the shunting inhibition and 39.8 Hz with the shunting inhibition (to compensate for the loss in input impedance in both branches).

 Parameters Figure S1.

For the simulation in panels D-F, 10 clusters of 10 excitatory (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons and Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) and 10 inhibitory (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) synapses where inserted at random, and within each cluster excitatory and inhibitory synapses where evenly spaced at 4 Math Eq {focus_keyword} Electrical Compartmentalization in Neuronsm intervals. Synapses were activated with Poisson spike trains of 3 Hz.

 Parameters Figure S2.

In panel D, synapse 1 was a non-plastic NMDA synapse (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) and synapse 2 a plastic synapse with the same parameters. When on, synapse 1 received a tonic spike train with a rate of 113 Hz (to yield a time-averaged activation Math Eq {focus_keyword} Electrical Compartmentalization in Neurons). Synapse 2 received rates ranging from 0 to 113 Hz, corresponding to the data points at different activations in the figure in the figure. For the simulations in panel H, we inserted NMDA synapses (Math Eq {focus_keyword} Electrical Compartmentalization in Neurons) in both branches and stimulated them with a rate of 200 Hz. Their maximal conductance Math Eq {focus_keyword} Electrical Compartmentalization in Neurons was optimized to obtain an average depolarization of −40 ± 2.5 mV in each branch.

 Data and Software Availability

Acknowledgments

Funding was received from the ETH Domain for the Blue Brain Project (BBP) and the European Union Seventh Framework Program (FP7/2007- 2013) under grant agreements FP7-26992115 (BrainScaleS) and FP7-604102 (The Human Brain Project), as well as EU grant agreement 720270 (HBP SGA1). We thank Profs. Idan Segev and Wulfram Gerstner and Dr. Arnd Roth for helpful advice and discussions and Drs. Luc Guyot, Jakob Jordan, and Aditya Gilra for proofreading the manuscript.

Author Contributions

W.A.M.W. and B.T.-N. designed the research. W.A.M.W. performed the mathematical modeling, implemented the models, and ran the simulations. W.A.M.W., B.T.-N., T.N., and M.-O.G. analyzed the results and wrote the paper.

Declaration of Interests

The authors declare no competing interests.

Supplemental Information

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Article Info

Publication History

Published: February 12, 2019

Accepted:
January 17,
2019

Received in revised form:
October 3,
2018

Received:
June 20,
2018

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DOI: https://doi.org/10.1016/j.celrep.2019.01.074

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© 2019 The Authors.

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