Parallel functional architectures within a single dendritic tree

SUMMARY The input-output transformation of individual neurons is a key building block of neural circuit dynamics. While previous models of this transformation vary widely in their complexity, they all describe the underlying functional architecture as unitary, such that each synaptic input makes a single contribution to the neuronal response. Here, we show that the input-output transformation of CA1 pyramidal cells is instead best captured by two distinct functional architectures operating in parallel. We used statistically principled methods to fit flexible, yet interpretable, models of the transformation of input spikes into the somatic “output” voltage and to automatically select among alternative functional architectures. With dendritic Na+ channels blocked, responses are accurately captured by a single static and global nonlinearity. In contrast, dendritic Na+-dependent integration requires a functional architecture with multiple dynamic nonlinearities and clustered connectivity. These two architectures incorporate distinct morphological and biophysical properties of the neuron and its synaptic organization.

-G, using a decoder on either the full (orange; v full ) or the subthreshold somatic membrane potential with (black; vsoma) or without (purple; v noNa ) dendritic Na + spikes. (B) Areas under curves (AUCs) for the ROCs shown in A. Box plots in B show median (horizontal line), 25th & 75th percentiles (box), ±1.5 inter-quartile ranges (whiskers), and outliers (circles) across 20 test trials. Data are from the 4-cluster arrangement. (B) Schematics of the different functional architectures used in the paper. Synapses on the biophysical model (top) are colored by the identity of the subunit processing their inputs in the cascade models (bottom). For the 'correct' (first, third, fifth) connectivities (used in Fig. 3 to Fig. 6), hot colors represent the subunits receiving the clustered excitatory inputs (and inhibitory inputs arriving on the same dendritic branches) while cold colors represent subunits receiving the background excitatory inputs (and inhibitory inputs arriving on the same dendritic branches). Inhibitory inputs targeting the soma of the biophysical model were placed on a separate leaf subunit (grey). Finally, the outputs of all leaf subunits were integrated by a single, linear subunit (cross) serving as the output of the cascade model. For the random connectivity ( Fig. 6B) synapses are assigned randomly to different subunits.  (B) Sample traces of the Na + differential voltage (v Na , black) and prediction error residuals of the dynamic cascade model and TCN (colors as in A) during the same trial as shown in A (black and red replotted from Fig. 3D). (C) Pearson correlation (R 2 ) and joint histogram of the dynamic cascade model's (left, replotted from Fig. 3E, left) and the TCN's vsoma prediction errors (right) and the Na + differential voltage, v Na . Data are from the 4-cluster arrangement. (D) Sample traces of the Na + differential voltage (v Na , black) and v Na predictions from the dynamic cascade model and TCN (colors as in A) during the same trial as shown in A (cf. Fig. 4A, bottom, black and red).  ). Numbers in parentheses on top of each sub-panel here and on the x-axes of all other panels show multiplicative factors used to change the dendritic NMDA and Na + maximal conductances, respectively, relative to the original parameters. (B) Cross-validated performance (variance explained) of the single-subunit static (gray) and multi-subunit dynamic cascades (red) for v noNa (left) and v Na (right) using different dendritic NMDA and Na + maximal conductances (numbers in parentheses as in A). Note that the Na + conductance multipliers are all 0 for v noNa as it is obtained by removing the effects of all dendritic Na + conductances. (C) Cross-validated performance (variance explained) of multi-subunit static (green), static with multiplexing (blue), and dynamic (red) cascade models for v noNa (left) and v Na (right) using the same alternative biophysical model parameterizations as in B. Data labelled as "original" is replotted from Fig. 4B. Box plots in B-C show median (horizontal line), 25th & 75th percentiles (box), ±1.5 inter-quartile ranges (whiskers), and outliers (circles) across 20 test trials. Data are from the 4-cluster arrangement.  Fig. 2A. Note the 4 smaller input clusters (pale colors) in addition to the larger clusters used for the simulation shown in the main text (bright colors). These smaller clusters receive inputs whose activity peaks outside of the somatic place field, which is around the middle of the track (as determined by the activity of inputs impinging the larger clusters). (B) Schematic showing the locations of input synapses along the dendritic tree as in Fig. 2B. Again, note the 4 smaller input clusters (pale colors) in addition to those used for the main simulations (bright colors). (C) Examples of simultaneous dendritic (pink and red, colors as in A-B) and somatic (black) membrane potential traces show dendritic Na + spikes and NMDA plateaus in the absence of somatic spiking. (D) Rate of spiking events (STAR Methods) in 34 basal dendritic branches (columns) and in the soma (rightmost column) at different spatial locations along the simulated maze (rows). Small and large colored circles along x-axis indicate cluster identity of corresponding branches receiving in-or out-of-field inputs, respectively (colors as in A-C).
(E) Histogram of the average event rate across dendritic branches. Gray, orange and cyan colors indicate branches receiving background, in-and out-of-field clustered inputs, respectively (see also legend in D for colors). (F) Histogram of the proportion of isolated dendritic events (i.e. without a simultaneous somatic action potential) across dendritic branches. Colors as in E. (G) Dendrite-soma coupling (STAR Methods; see Rolotti et al. (2022) [S4]) as a function of the distance of the branch from the soma. Colors as in E. (B) Cross-validated performance (variance explained) of the single-subunit static (gray) and multi-subunit dynamic cascade models (red) for v noNa (left) and v Na (right) with original inputs (replotted from Fig. S6B) and with out-of-place field clustered inputs (see also Fig. S7) . (C) Cross-validated performance (variance explained) of multi-subunit static (green), static with multiplexing (blue), and dynamic (red) cascade models for v noNa (left) and v Na (right) with original (replotted from Fig. 4B) and out-of-place field clustered inputs (see also Fig. S7). Box plots in B-C show median (horizontal line), 25th & 75th percentiles (box), ±1.5 inter-quartile ranges (whiskers), and outliers (circles) across 20 test trials. Data are from the 4-cluster arrangement (original) with an additional 4 out-of-place field clusters for the new simulations. Fig. 4. (A) Top: Spike triggered average difference (thick black curve) between the biophysical model's somatic membrane potential with action potentials (v full ) and without somatic Na + channels (vsoma). Grey lines show individual traces. Bottom: Pearson correlation (R 2 ) and joint histograms of v full (x-axis) and vsoma + average action potential from top panel (y-axis). A high correlation indicates a simple, stereotypical additive contribution of somatic action potential generation to the somatic membrane potential. (B) Similar to panel A for 4 different dendritic branches (inset on left), showing somatic spike-triggered bAPs (top) and their contributions to the local dendritic membrane potential. A high correlation in bottom panels indicates a simple, stereotypical additive contribution of bAPs to the local dendritic membrane potential, and thus no discernible effect on dendritic integration. (C) Pearson correlation (R 2 ) and joint histograms of the the dynamic cascade model's (inset on left) subunit outputs for predicting v noNa (top) and v Na (bottom) with (x-axis, i.e. training on v full ) and without incorporating bAPs (y-axis, i.e. training on vsoma). Data are from the 4-cluster arrangement with out-of-place field clustered inputs (Fig. S7). Inset shows schematic architectures of cascade models: static, static with multiplexing, and dynamic cascades used throughout the paper, in which subunits independently feed into the output (respectively green, blue, and red as in the inset of Fig. 4A), and a dynamic model that can incorporate arbitrary interactions between subunits (purple). Although the dynamical model with interactions was implemented technically as a single subunit, it was fundamentally more flexible than the single-unit dynamical architecture shown in Fig. S4 ("dynamic: global", purple): the former received the spike trains of each subunit as a separate input (thus retaining the notion of 'subunits'), while the latter received a single input which combined all input spike trains (see also STAR Methods).  Table S1. Summary of differences between v noNa and v Na architectures. Related to Fig. 7.

Figure S9. Effects of backpropagating action potentials (bAPSs) on dendritic integration. Related to
S8C,E S8D,F S10A-E S11A-E S12A S12C-D Table S2. Summary of biophysical and cascade models used in figures. Related to STAR Methods.
indicates figure panels that have the indicated models' somatic membrane potential output explicitly plotted. indicates panels that have the biophysical models' dendritic membrane potential explicitly plotted. indicates panels that use the indicated models' somatic outputs for various calculations other than for assessing model fits (e.g. spike prediction analysis).
indicates panels that use the indicated models' subunit outputs for various calculations other than for assessing model fits (e.g. spike prediction analysis). indicates panels that use the indicated models' outputs specifically for assessing model fits (e.g. variance explained).
indicates panels that use cascade models that were fitted to the indicated voltage target. Lighter versions of each check represent instances where the models' dendritic voltage traces were used rather than their somatic outputs. v full is the somatic membrane potential including action potientials. vsoma is the subthreshold somatic membrane potential with somatic and axonal Na + channel conductance set to 0 in the biophysical model. v noNa is the subthreshold somatic membrane potential in the absence of axonal, somatic and dendritic Na + channels. v AP = v full -vsoma and v Na = vsoma-v noNa . TP (true positive) P(prediction=1, true=1) FN (false negative) P(prediction=0, true=1) FP (false positive) P(prediction=1, true=0) TN (true negative) P(prediction=0, true=0) hit rate (recall) P(prediction=1 | true=1) = TP / (TP + FN) false alarm rate P(prediction=1 | true=0) = FP / (TN + FP) precision P(true=1 | prediction=1) = TP / (TP + FP) Table S3. Spike prediction accuracy measures. Related to STAR Methods.