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 impedancebased 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 contextdependent manner. We also find that this dynamic recompartmentalization can enable branchspecific learning of stimulus features. Analysis of dendritic patchclamp recording experiments confirmed our theoretical predictions.
Graphical Abstract
Keywords
 neural computation
 dendrites
 compartmentalization
 independent subunits
 branchspecific 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 Nmethyl Daspartate (NMDA), Ca^{2+}, 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 voltagedependent ionchannel 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.
 Mel B.
 Schiller J.
Encoding and decoding bursts by NMDA spikes in basal dendrites of layer 5 pyramidal neurons.
) and hence, through branchspecific plasticity (
Golding et al., 2002
 Golding N.L.
 Staff N.P.
 Spruston N.
Dendritic spikes as a mechanism for cooperative longterm potentiation.
,
Govindarajan et al., 2011
 Govindarajan A.
 Israely I.
 Huang S.Y.
 Tonegawa S.
The dendritic branch is the preferred integrative unit for protein synthesisdependent 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.
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Locationdependent synaptic plasticity rules by dendritic spine cooperativity.
), drive the clustering of correlated synaptic inputs (
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.
,
,
Lee et al., 2016
 Lee W.C.A.
 Bonin V.
 Reed M.
 Graham B.J.
 Hood G.
 Glattfelder K.
 Reid R.C.
Anatomy and function of an excitatory network in the visual cortex.
). A recent finding that distal apical dendrites can spike 10fold more often than somata (
Moore et al., 2017
 Moore J.J.
 Ravassard P.M.
 Ho D.
 Acharya L.
 Kees A.L.
 Vuong C.
 Mehta M.R.
Dynamics of cortical dendritic membrane potential and spikes in freely behaving rats.
) suggests an important role for this branchspecific plasticity. Independent subunits furthermore allow different input streams to be discriminated from each other (
Johenning et al., 2009
 Johenning F.W.
 Beed P.S.
 Trimbuch T.
 Bendels M.H.K.
 Winterer J.
 Schmitz D.
Dendritic compartment and neuronal output mode determine pathwayspecific longterm potentiation in the piriform cortex.
), and they facilitate sensory perception through feedback signals (
Takahashi et al., 2016
 Takahashi N.
 Oertner T.G.
 Hegemann P.
 Larkum M.E.
Active cortical dendrites modulate perception.
).
When triggered independently, these local regenerative events are predicted to enable individual neurons to function as twolayer neural networks (
,
Poirazi et al., 2003b
 Poirazi P.
 Brannon T.
 Mel B.W.
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
 Schiess M.
 Urbanczik R.
 Senn W.
Somatodendritic synaptic plasticity and errorbackpropagation in active dendrites.
) and implement translation invariance (
Mel et al., 1998
 Mel B.W.
 Ruderman D.L.
 Archie K.A.
Translationinvariant 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.
 Mel B.W.
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
 Bono J.
 Clopath C.
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.
 Morrison A.
Effects of calcium spikes in the layer 5 pyramidal neuron on coincidence detection and activity propagation.
,
Larkum et al., 1999
 Larkum M.E.
 Zhu J.J.
 Sakmann B.
A new cellular mechanism for coupling inputs arriving at different cortical layers.
), and support the backprop algorithm to train neural networks (
Guerguiev et al., 2017
 Guerguiev J.
 Lillicrap T.P.
 Richards B.A.
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 impedancebased 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 impedancebased independence index . We compute this index from dendritic patchclamp 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 patchclamp 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 impedancebased 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 branchspecific 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 (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.
 Stevens C.F.
A quantitative description of NMDA receptorchannel kinetic behavior.
,
MacDonald and Wojtowicz, 1982
 MacDonald J.F.
 Wojtowicz J.M.
The effects of Lglutamate and its analogues upon the membrane conductance of central murine neurones in culture.
). If 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, contextdependent 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).
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
 Cuntz H.
 Forstner F.
 Borst A.
 Häusser M.
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 describes how an input current at location along the dendrite will change the voltage at any location . In prior work, the values of this matrix (termed input impedances when and transfer impedances when ) were used to shed light on the processing properties of various cell types (
Koch et al., 1982
 Koch C.
 Poggio T.
 Torre V.
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
 Koch C.
 Douglas R.
 Wehmeier U.
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 . 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 steadystate 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 depthfirst manner (
; Figures S1A–S1C), supports a treelike 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 depthfirst 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 ( to 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 morphoelectric transform by
Zador et al., 1995
 Zador A.M.
 AgmonSnir H.
 Segev I.
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 ) 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 αamino3hydroxy5methyl4isoxazolepropionic acid (AMPA) synapses, it is straightforward to use other voltage dependent currents (i.e., NMDA or voltagegated 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 voltagedependent 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
 Wang Y.
 Gupta A.
 ToledoRodriguez M.
 Wu C.Z.
 Markram H.
Anatomical, physiological, molecular and circuit properties of nest basket cells in the developing somatosensory cortex.
], a granule cell [
CarimTodd et al., 2009
 CarimTodd 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.
], and a L5 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.
]; 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, intradendritic dynamics and regenerative NMDA events are well reproduced by NETs (Figures S1E and S1F).
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 and do not differ too much. If region 1 would, for instance, be much closer to a bifurcation than region 2, so that , 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 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 (Figure 2G). We nevertheless wondered whether it is possible to find a threshold 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 twophoton glutamate uncaging;
Pettit et al., 1997
 Pettit D.L.
 Wang S.S.H.
 Gee K.R.
 Augustine G.J.
Chemical twophoton uncaging: a novel approach to mapping glutamate receptors.
) leads to a marked attenuation of the measured signal (typically Ca^{2+} 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 Ca^{2+} signal itself is thought to be more local than the voltage signal (
Biess et al., 2011
 Biess A.
 Korkotian E.
 Holcman D.
Barriers to diffusion in dendrites and estimation of calcium spread following synaptic inputs.
,
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.
,
Simon and Llinás, 1985
 Simon S.M.
 Llinás R.R.
Compartmentalization of the submembrane calcium activity during calcium influx and its significance in transmitter release.
,
Yuste and Denk, 1995
 Yuste R.
 Denk W.
Dendritic spines as basic functional units of neuronal integration.
) while also being influenced by the nonlinear activation function of voltagedependent calcium channels (
Almog and Korngreen, 2009
 Almog M.
 Korngreen A.
Characterization of voltagegated 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 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 suprathreshold 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 , 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 , so that for , this correlation is still close to 1 (Figure 2G).
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 (Equation 2), the sum of all voltage components associated with nodes that integrated more than one region) in the synaptic currents by their longterm average (here computed as a 200 ms lowpass 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 ; for , the shape of the NET voltage trace at leaf 1 is drastically modified, whereas at , 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 , 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 , the iNET and NET dynamics agree very well (Figure 4C, bottom). Analyzing the correlations between both leaf voltage components confirms this finding; for , the NET correlation approaches the iNET value of zero (Figure 4D). The RMSE between the somatic traces also vanishes for . 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 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 , pairs of dendritic regions can be considered independent.
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 values a fully developed NMDA spike can form independently in a branch (Figure S2B). We also asked at which 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 (Figures S2D and S2E). These analyses corroborated our conclusion that, for 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 (see STAR Methods). For the stellate and granule cells, we did not identify any compartments, as there were no values greater than 10 in these cells (Figures 4H and 4I). In the pyramidal cell, the number of compartments for was far less than the number of dendritic terminals (Figure 4J). Even at lower values, there are still much fewer independent subunits than dendritic terminals (Figure S3; see, for instance, compartment numbers for ). 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 values, and thus the degree of electrical compartmentalization, are physiologically plausible, we reanalyzed dual dendritic patchclamp 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 patchclamp recording study.
). We computed 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 steadystate 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 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 predicted by the models is correct) are compatible with experimental data.
Dendritic Compartmentalization Can Be Modified Dynamically by Inputs
As neurons perform different inputoutput transformations at different moments in time, such as during up and downstates (
Wilson and Kawaguchi, 1996
 Wilson C.J.
 Kawaguchi Y.
The origins of twostate 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.
Branchspecific dendritic Ca(2+) spikes cause persistent synaptic plasticity.
,
PolegPolsky et al., 2018
 PolegPolsky 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 , a shunting conductance , and excitatory conductances :
(Equation 5)
can be reinterpreted in the following way (see Methods S1):
(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 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 highconductance state (
Destexhe et al., 2003
 Destexhe A.
 Rudolph M.
 Paré D.
The highconductance 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 timeaverage 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 highconductance NET is accurate in reproducing the average voltages; traces in Figure 6D (full colored lines) agree very well with the average postsynaptic potentials computed during an ongoing barrage of balanced excitation and inhibition (dashed colored lines).
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 branchselective 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 timeaveraged conductance of 5 nS) could be provided by the somatostatinpositive 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.
 AlonsoNanclares 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.
Layerspecific modulation of neocortical dendritic inhibition during active wakefulness.
). This change in compartmentalization can be quantified by the change in (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 .
Dynamic Compartmentalization Can Enable BranchSpecific Learning
Recent experiments have demonstrated that inhibitory interneurons are required for branchspecific plasticity (
Cichon and Gan, 2015
 Cichon J.
 Gan W.B.
Branchspecific dendritic Ca(2+) spikes cause persistent synaptic plasticity.
). Can a transient recompartmentalization, mediated by inhibition, underlie this branchspecific learning? Before learning, postsynaptic 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 submillisecond 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 backpropagating action potentials during induction of associative longterm 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 (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 from 3.0 to 8.5 (Figure 7C, black versus blue tree). The original 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 , thus anticorrelating the weight evolution (Figure 7F) and enabling branchspecific learning (Figures 7E and 7G).
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 as a ratio of these impedances and have shown that this number is a good and measurable predictor of independence. We found that for , pairs of dendritic sites function as independent subunits. Furthermore, we have used the NET to design an algorithm that, given a threshold , yields the maximal number of regions that can coexist on the dendritic tree separated by at least this threshold .
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 vivolike conditions.
), that many branches are not separated by 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 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 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 , triple dendritic recordings of parent and higherorder 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 Ca^{2+} 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 synthesisdependent 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.
Locationdependent 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 twostate 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.
Glutamatebound NMDARs arising from in vivolike network activity extend spatiotemporal 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 highconductance 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 NMDAspike 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 NMDAspike 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 between and , and in smaller cells, these numbers increase 5fold (Figure S3). This suggests that global conductance input increases the number of compartments 2 to 5fold, depending on the cell’s properties. Inspecting pairwise independence, we determine that in pyramidal cells, 5% to 10% of terminal pairs are separated by 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 voltagebased plasticity rule (
Clopath et al., 2010
 Clopath C.
 Büsing L.
 Vasilaki E.
 Gerstner W.
Connectivity reflects coding: a model of voltagebased STDP with homeostasis.
) and explored whether these synapses could learn independently. Consistent with our prior observations, we found that a value of was too small for these branches to learn independently. Nevertheless, shunting inhibition that increased to 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.
Branchspecific 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 timescales, 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 patchclamp 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; Caspike 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 branchselective integration in CA1 pyramidal cells.
), and the importance of interneuron activity for branchspecific learning (
Cichon and Gan, 2015
 Cichon J.
 Gan W.B.
Branchspecific dendritic Ca(2+) spikes cause persistent synaptic plasticity.
) suggest that dynamic compartmentalization is ubiquitous in normal brain function, with farreaching 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 wellknown 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 TorbenNielsen (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.
 ToledoRodriguez 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 (
CarimTodd et al., 2009
 CarimTodd 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.
 AlonsoNanclares 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 , the membrane conductance , the capacitance and the intracellular resistance .
To generate somatic APs, we used the fast inactivating current and the fast, noninactivating current 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: and . The leak current was then fitted to yield a membrane timescale of and an equilibrium potential of .
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 timeinvariant 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.
):
(Equation 7)
For AMPA synapses, we used rise resp. decay times , for the conductance window and a reversal potential , while for GABA synapses we used , and . For NmethylDaspartate (NMDA) channels (
Jahr and Stevens, 1990a
 Jahr C.E.
 Stevens C.F.
A quantitative description of NMDA receptorchannel kinetic behavior.
,
MacDonald and Wojtowicz, 1982
 MacDonald J.F.
 Wojtowicz J.M.
The effects of Lglutamate and its analogues upon the membrane conductance of central murine neurones in culture.
), the synaptic current had the form
(Equation 8)
and the rise resp. decay time were , , and , while 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:
(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 NMDAactivated macroscopic conductances predicted by singlechannel kinetics.
:
(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.
):
(Equation 11)
In the remainder of this work, we will refer to the voltagedependent factors in the synaptic input current as the ‘synaptic voltage dependence’ (SVD), denoted by . Hence, for AMPA or GABA synapses
(Equation 12)
and for NMDA synapses
(Equation 13)
Note that when we refer to the conductance of a simple synapse, we mean the maximum value 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 , 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 voltagebased STDP with homeostasis.
) where the evolution of the weight of a given synapse depends both on the postsynaptic voltage and the presynaptic AP inputs. This leads to a synaptic current of the following form:
(Equation 14)
In all our simulations the initial weight was 1 and, during the simulation, the weight could fluctuate in the interval .
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.
 TorbenNielsen B.
The Green’s function formalism as a bridge between single and multicompartmental modeling.
,
Wybo et al., 2015
 Wybo W.A.M.
 Boccalini D.
 TorbenNielsen B.
 Gewaltig M.O.
A sparse reformulation of the Green’s function formalism allows efficient simulations of morphological neuron models.
) . The GF is a function of three variables: two locations and along the dendritic arborization and a temporal variable . We compute the GF in an exponential basis:
(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 . 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 , to highlight the difference between the now discrete indices and and the continuous variable .
To compute the output voltage at location for a given input current at location , one needs to compute the convolution of the GF evaluated at and with this input current (
Koch, 1998
 Koch C.
Biophysics of Computation: Information Processing in Single Neurons (Computational Neuroscience).
):
(Equation 16)
for which we will use the shorthand
(Equation 17)
Since 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:
(Equation 18)
In the rest of the text, it will be understood that without temporal coordinate refers to the steady state impedance – which we will simply call ‘the impedance’ for brevity – while 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 , where it is implied that both and are time dependent quantities since is the temporal impedance kernel. Conversely, writing means that is the steady state impedance value, and thus and 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 (M).
Synaptic activation
The eventual steady state voltage obtained after activating a synaptic conductance at location depends for a large part on the input impedance . Following (17), it can be obtained as a solution of the equation
(Equation 19)
This solution is thus a function of the product of impedance and conductance:
(Equation 20)
We refer to this product as the synaptic activation 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 depthfirst 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 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 in Figure 1A to , the pink and green nodes both integrate the identical subset . 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 – in analogy to the average error in classical biophysical models, which depends on . 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 M (Figure S4B). M is thus a good choice for .
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 , many of high , and relatively few of intermediate . 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 above this boundary will belong to the proximal domain (node in Figure S4E), whereas connected regions with below this boundary will constitute distal domains (colored red in Figure S4E, node ). 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 .
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 intradendritic synaptic interactions. Mathematically, their contribution to the somatic voltage can be written as:
(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:
(Equation 23)
This current will become truly independent from all other synapses if
(Equation 24)
with a (large) number that has to be determined empirically and denoting the short term fluctuations of around a long term average . Here, short term means the timescale on which neurons convert electrical inputs to output. As NMDA synapses have a decay time constant of ms, this averaging timescale should be at least somewhat larger than the NMDA time constant (we chose ms). Then (23) can be approximated as follows:
(Equation 25)
The longterm 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:
(Equation 26)
Consequently, the solution for 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:
(Equation 27)
where denotes the number of regions node integrates. When we are interested in determining whether a pair of regions (integrated by the leafs , ) 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 (i.e., a sum over impedances of nodes that integrate one region but not the other) and a root impedance (i.e., a sum over impedances of nodes that integrate both regions). Then, regions are independent if:
(Equation 28)
For mutual independence between and , this equation has to hold for both and . To summarize these two conditions in a single expression, we defined the ‘ impedancebased independence index’ :
(Equation 29)
Then, if (28) holds for both regions, the following condition also holds:
(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 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 above a certain threshold, will be called a compartmentalization of that dendritic tree for that given . 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 , 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 in the NET tree forms a valid compartment, all nodes in the subtree of are part of the same compartment, since their to other compartments will be higher than the of . 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 on the path from N to the root. We check whether the following condition holds
(Equation 31)

2.
with the path from 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 , these nodes will be separated by at least the required . Hence, we say that is a tentative compartment with respect to .

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 and its child leafs. Then, if at least two child leafs are tentative compartments with respect to , the other leafs are removed. Otherwise, all child leafs but the one with largest impedance are removed. Note that in the latter case, 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 . This node will be a compartment node in the final compartmentalization.
Extracting I_{Z} from patchclamp data
Extrapolating from twoelectrode recordings
Although there was only data available from twoelectrode patch clamp setups with one dendritic and one somatic electrode, it is still possible to extrapolate what the value of would be between two dendritic locations, if the second location would be on a branch with similar electrical properties and at a similar distance from the soma as the first location . From the dendritic current injection, we can calculate the input impedance and the dendritetosoma transfer impedance by fitting a regression line to the IVcurve. Conversely, from the somatic current injection, we can calculate the somatic input impedance as well as the somatodendrite transfer impedance in a similar manner. We then assume that , and . between these two sites is given by:
(Equation 32)
with , and 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).
):
(Equation 33)
Estimating the dendritic access resistance
Full compensation of wholecell 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.
):
(Equation 34)
to the voltage transient after stimulus onset. To do so, we minimized a weighted sum of squares error
(Equation 35)
where the weights were given by (with ms), a function chosen to prioritize accuracy at small values (Figures S5A and S5B). The resulting fit contained one fast and one slow timescale (let us index the fast timescale with 1), and we equated the superfluous voltage drop due to the access resistance to the prefactor of the fastest exponential:
(Equation 36)
It can be seen that this analysis yielded a colinear 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 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 bridgebalance 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).
Simulationspecific parameters
Parameters Figure 1.
We evaluated the impedance matrix in panel B at 10 m intervals. In the simulation depicted in panels J and K, the synapses contained only an AMPA component with and no active channels were inserted in the soma.
Parameters Figure 2.
100 NMDA ( and ) and 100 GABA () 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 and . 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 CD, NMDA synapses () were used. Their and incoming Poisson rate were optimized to utilize the full range of the NMDA nonlinearity. For the simulations in panel E, NMDA ( and ) and GABA () synapses were used and their rates were also optimized to utilize the full range of the NMDA nonlinearity.
Parameters Figure 6.
For the simulations in panel C, the main synapses contained only an AMPA component with . To simulate the highconductance state, 200 AMPA and 200 GABA synapses () 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 timeaveraged conductances of all background synapses were inserted in the morphology as static shunts.
The inhibitory synapse in panels EH had and was activated at a steady rate of 200 Hz, so that it’s total timeaveraged conductance was around 5 nS. For the simulations in panel H, we inserted NMDA synapses () 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 of the NMDA synapse was optimized to obtain an average depolarization of in each branch, a target value which yields parameters that allow exploitation of the full range of the NMDA nonlinearity.
Parameters Figure 7.
In both simulations with and without shunting inhibition, noise was implemented at all three locations using AMPA () and GABA () 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 () receiving a Poisson train with a rate of 277 Hz (tuned to reach a timeaveraged conductance of 12 nS) during the 100 ms learning intervals. Note that this single conductance could again represent multiple synapses.
Stimulusspecific innervation patterns were: to : 5 synapses, to : 2 synapses, to : 2 synapses and to : 5 synapses. These synapses were all NMDA synapses () 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 DF, 10 clusters of 10 excitatory ( and ) and 10 inhibitory () synapses where inserted at random, and within each cluster excitatory and inhibitory synapses where evenly spaced at 4 m intervals. Synapses were activated with Poisson spike trains of 3 Hz.
Parameters Figure S2.
In panel D, synapse 1 was a nonplastic NMDA synapse () 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 timeaveraged activation ). 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 () in both branches and stimulated them with a rate of 200 Hz. Their maximal conductance 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 FP726992115 (BrainScaleS) and FP7604102 (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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