An efficient analytical reduction of detailed nonlinear neuron models

Abstract

Detailed conductance-based nonlinear neuron models consisting of thousands of synapses are key for understanding of the computational properties of single neurons and large neuronal networks, and for interpreting experimental results. Simulations of these models are computationally expensive, considerably curtailing their utility. Neuron_Reduce is a new analytical approach to reduce the morphological complexity and computational time of nonlinear neuron models. Synapses and active membrane channels are mapped to the reduced model preserving their transfer impedance to the soma; synapses with identical transfer impedance are merged into one NEURON process still retaining their individual activation times. Neuron_Reduce accelerates the simulations by 40-250 folds for a variety of cell types and realistic number (10,000-100,000) of synapses while closely replicating voltage dynamics and specific dendritic computations. The reduced neuron-models will enable realistic simulations of neural networks at unprecedented scale, including networks emerging from micro-connectomics efforts and biologically-inspired "deep networks". Neuron_Reduce is publicly available and is straightforward to implement.

Fig. 1. An analytic method for reducing neuron model complexity (Neuron_Reduce).

a Detailed passive model of 3D reconstructed L5 thick-tufted pyramidal cell from rat neocortex. Its nine stem dendrites (one apical and 8 basal) are depicted in different colors. b Each original stem dendrite is reduced to a single cylinder that retains the specific passive cable properties (R_m, C_m, and R_a) of the original tree. The diameter and length of the respective cylinders are computed analytically using Eqs. (1)–(11), such that each cylinder preserves both the transfer resistance from the most electrotonically distal dendritic tip to the soma as well as the input resistance at the soma end of the corresponding stem dendrite. This generates a unique cylindrical cable for each of the original stem dendrites. Scale bars in a, b are 100 µm. c Synapses with similar transfer resistance to the soma (exemplar synapses are marked as 1–4 at top right) are all mapped to the respective locus in the reduced cylinder so that their transfer resistance is similar in the two models. In the reduced model, these synapses are merged into one “NEURON” process (red synapse in b), but they retain their individual activation time (see Methods and Supplementary Fig. 1). The same mapping also holds for active membrane conductances (yellow region, denoting the Ca²⁺ “hot spot” in the apical tree). d Transfer impedance $\left(Z_{d,0}=Z_{0,d}\right)$ between point d on the apical tree (shown in a, b) and the soma (X = 0) as a function of the input frequency in both the detailed (black trace) and the reduced (red trace) models. e Composite somatic EPSPs resulting from sequential activation of the four distal apical synapses shown in c in the detailed model (black trace) and the reduced model (red trace). In this simulation the dendritic tree was passive. The synapses were activated in temporal order 1, 2, 3, 4 as shown by the vertical lines below the composite EPSP. The respective peak conductances of these AMPA-based synapses were 0.6, 0.3, 0.4, and 0.4 nS (details in Supplementary Table 2 and see Supplementary Fig. 1 for the active case).

Fig. 2. Neuron_Reduce faithfully replicated the I/O properties of a detailed nonlinear model of a L5 pyramidal cell.

a Layer 5 pyramidal cell model as in Fig. 1a, with 8000 (AMPA + NMDA) excitatory (magenta dots) and 2000 inhibitory synapses (cyan dots, see Supplementary Table 2 for synaptic parameters). Excitatory synapses were activated randomly at 5 Hz and the inhibitory synapses at 10 Hz. This detailed model consists of a dendritic Ca²⁺ “hot spot” (as in Fig. 1c) and a Na⁺ spiking mechanism at the cell body. Scale bar 100 µm. b An example of the voltage dynamics at the soma of the detailed model (black trace) and the reduced model (red trace); spike times are represented by the black and red dots above the respective spikes. c Cross-correlation between spikes in the reduced versus the detailed models. d Inter-spike interval (ISI) distributions for the two models. e Output firing rate of the reduced (red) versus the detailed (black) models as a function of the firing rate of the excitatory synapses. Gray dots represent the case shown in b. f SPIKE-synchronization measure between the two models as a function of the firing rate of the detailed model for the case of only AMPA (blue) and AMPA + NMDA synapses (orange). The performance of the reduced model with NMDA synapses was lower for low output frequency, but improved significantly for output frequencies above ~7 Hz (see Discussion). g SPIKE synchronization between the detailed and the reduced models as a function of the firing rate of the detailed model, for active and passive dendrites, and with/without NMDA-based synaptic conductance.

Fig. 3. Neuron_Reduce enhances the simulation speed by up to several hundred fold.

a Simulation run-time for the detailed (black) and the reduced models (red) of layer 5 pyramidal cell shown in Fig. 2a, for a simulation of 50 s, and their ratio (the speed-up, gray) as a function of the number of simulated (GABAA-, AMPA- and NMDA- based) synapses. Due to the almost constant run-time of the reduced model, the run-time ratio increases with larger number of synapses. Above 75,000 synapses, an additional effect becomes visible: the detailed model no longer fits into the cache of the CPU and exhibits a supralinear increase in run-time. This can be seen by the black curve deviating from the dotted red curve, which shows the expected simulation time for the detailed model assuming a constant computation cost per synapse (see also Supplementary Table 1). b Accuracy (blue) of the reduced model and its speed-up in simulation run-time (gray) as a function of the number of electrical compartments per length constant for a neuron with 10,000 synapses (50 s per simulation).

Fig. 4. The dendritic potential in the reduced model represents the average dendritic voltage dynamics in the detailed model.

a Detailed model (left) and reduced model (right) of the cell shown in Fig. 2. Dendritic branches of the same color in the detailed model are all mapped to the respective compartment with identical color in the reduced model. b For each of the four colored regions shown in a (and respective colored sphere at top left), the voltage transients in individual branches are shown by the gray traces. Superimposed in black is the average voltage of these traces and in red is the voltage transient in the respective compartment in the reduced model. The somatic spikes in the detailed model (black) and reduced model (red) are also shown. The simulation is as in Fig. 2e, with excitatory synapses firing at 5.5 Hz. Scale bars for the respective morphologies are 100 µm.

Fig. 5. Dendritic Ca²⁺ spike and BAC firing faithfully replicated in the reduced model.

a, b (Left) Detailed L5 pyramidal cell model with nonlinear Ca²⁺ “hot spot” (same model as in Fig. 2). a Injecting a depolarizing step current to the soma (0.95 nA for 8.5 ms) in the detailed model evoked a somatic action potential, AP (black trace) that propagated backward semi-actively into the apical tree (red trace). b Combining the somatic input with a transient synaptic-like current injection (0.95 nA peak value with 0.5 and 5 ms rise time and decay time, respectively; red transient) to the “hot region” in the apical dendrite evoked a prolonged local Ca²⁺ spike, which, in turn, triggered a burst of two extra somatic Na⁺ spikes (the BAC firing phenomenon). c, d Same as in a, b, but for the reduced model. Scale bars for the respective morphologies are 100 µm.

Fig. 6. Discriminating spatio-temporal input sequences in the detailed versus the reduced model.

a A model of L5PC (detailed model, Fig. 1) with 12 excitatory synapses spatially distributed on one of its basal dendrites (red dots on green basal dendrite). b Somatic responses to sequential activations of its basal synapses in the IN (cyan) and the OUT (blue) directions. In this case, the synaptic model only consists of an AMPA component. c As in b but the synaptic model consists of both AMPA and NMDA components. d Reduced model for the detailed model shown in a. Neuron_Reduce mapped the 12 synapses in the detailed model into five synapses in the reduced model. e, f. As in b, c, but for the reduced model. g Pattern separability (see Methods) of the detailed (black) and the reduced (red) models when the synaptic model only consists of an AMPA component. h As in g, after subtracting the peak voltage obtained in the OUT direction from each of the voltage responses. i, j As in g, h but when the synaptic models consisted of both AMPA and NMDA conductances. Note the similarity between the detailed and the reduced models in terms of pattern separability.

Fig. 7. Neuron_Reduce working successfully on a variety of neuron models.

a–c Detailed models of three somatosensory neurons (left, L6 tufted pyramidal cell in green; middle, L2/3 large basket cell in red; and right, L4 double bouquet cell in blue) and their respective reduced models. Scale bars 100 µm. d–f Voltage responses to an excitatory synaptic input activated at 1.8, 2.9, and 3.17 Hz, respectively, for both the detailed (black) and the reduced models (corresponding colors). The inhibitory input activation rate was 10 Hz for all models. g The SPIKE-synchronization index for the 13 detailed versus reduced neuron models. The mean simulation speed-up for the L6 tufted pyramidal cell, L5 Martinotti cell, and L4 spiny stellate cell were 95, 40, and 60, respectively. See Supplementary Table 2 for cell models and input parameters and Supplementary Fig. 6 for the SPIKE-synchronization measure on additional 88 modeled cells.