High-capacity embedding of synfire chains in a cortical network model

Abstract

Synfire chains, sequences of pools linked by feedforward connections, support the propagation of precisely timed spike sequences, or synfire waves. An important question remains, how synfire chains can efficiently be embedded in cortical architecture. We present a model of synfire chain embedding in a cortical scale recurrent network using conductance-based synapses, balanced chains, and variable transmission delays. The network attains substantially higher embedding capacities than previous spiking neuron models and allows all its connections to be used for embedding. The number of waves in the model is regulated by recurrent background noise. We computationally explore the embedding capacity limit, and use a mean field analysis to describe the equilibrium state. Simulations confirm the mean field analysis over broad ranges of pool sizes and connectivity levels; the number of pools embedded in the system trades off against the firing rate and the number of waves. An optimal inhibition level balances the conflicting requirements of stable synfire propagation and limited response to background noise. A simplified analysis shows that the present conductance-based synapses achieve higher contrast between the responses to synfire input and background noise compared to current-based synapses, while regulation of wave numbers is traced to the use of variable transmission delays.

Fig. 1

(a) Construction of synfire chain embedding in a population of neurons. A sequence of excitatory pools (green ellipses) is formed by randomly selecting n _E distinct neurons from the excitatory population (large green circle) with replacement to form each pool; two such selections are indicated in brown. A corresponding sequence of inhibitory pools (red) is formed from the inhibitory population (red). Each neuron appears in many pools; e.g. each of the neurons in blue appears in two of the pools shown. Links consist of all-to-all connections from each excitatory pool to the next excitatory pool as well as to the corresponding inhibitory pool (arrows). (b) Origin of excitatory background input. Background input to neuron i arises from the activity of the n _E neurons in each of its C _E/n _E predecessor pools. Each neuron j in a predecessor pool of i fires at rate ν _S + ν _W, with the component ν _W being due to spike packets in all C _E/n _E pools that j belong to. From this must be subtracted (n _E/C _E)ν _W, the rate due to spike packets in the predecessor pool of i. This relationship is formalized in Eq. (4)

Fig. 2

Wave activity and mean spiking rate in networks with ; (a-d) small pools, n _E = 72; (e–h) large pools, n _E = 200 (a,e) Spike rasters of neurons in (a) pools 20 to 79, (e) pools 320 to 419 in the sequence beginning with the stimulated pool (left axis index). For each pool, ten arbitrarily chosen neurons are shown (indexed cumulatively on the right axis); (b,f) raster of extracted spike packets, rectangles indicate regions shown in (a,e); (c,g) number of waves; (d,h) population mean rate histograms

Fig. 3

Dependence of wave propagation and stochastic spiking on background input rate, λ _E. (a) probability of wave propagation over 98 pools, P _S(λ _E,γλ _E), for λ _E ∈ [0,300] kHz and pool sizes n _E ∈ 60,68,...,220; λ _E,max: P _S(λ _E,max,γλ _E,max) = 0.5 increases with n _E. (b) probability that each neuron in a pool participates in a spike packet, p _f, and (c) mean pool-to-pool propagation time of a wave, T, for n _E ∈ 60,68,...,220; in (b,c) each curve is defined on for λ _E: P _S(λ _E,γλ _E) > 0, the upper bound of which increases with n _E. (d) maximum background input rate for propagation versus pool size, λ _E,max(n _E). (e) stochastic spiking rate ν _S = f _S(λ _E,γλ _E) for three values of inhibitory synaptic conductance, g _I. (f) probability of wave survival over 98 pools versus equilibrium number of waves when driven by external wave stimulation at rate 25 Hz for n _E = 80 (dotted), n _E = 200 (dashed). (See text, Section 3.3)

Fig. 4

(a–c) Predictions of mean field analysis for embedding level (α = p/N _E), mean firing rate (ν) and equilibrium number of synfire waves per neuron (h _eq/N _E) as functions of pool size and connectivity. (d) comparison between mean field analysis and simulations for connectivity . For each C _E: upper plot, overall mean rate and mean rate due to wave spiking versus pool size for the mean field solution (small green markers) and for simulations (large magenta markers); lower plot, versus pool size, the mean field prediction for the equilibrium number of waves (small green markers) and the maximum and mean numbers of waves in the simulations (large magenta triangles and circles, respectively)

Fig. 5

Measures of embedding capacity: (a) curves in (n _E,C _E)-space for four equilibrium firing rates (ν _eq =1,2,5,10 Hz) and for two upper limits (C _E,max1(n _E) for global rate stability and C _E,max2(n _E) for ν _W/ν > 0.5) and the points of intersections (with C _E,max1(n _E) red ‘o’s, with C _E,max2(n _E) green ‘o’s); (b) corresponding embedding capacity α _max(C _E) curves and limits, simulation results (‘x’); (c,d) as for (a,b) but for

Fig. 6

Dependence of embedding capacity on inhibitory conductance. Embedding capacity curves α _max(C _E) for 3 equilibrium firing rates ν _eq = 2, 5, 10 Hz (panels a–c); for g _I = 0.1 (red), 0.11, (magenta), 0.12 (blue). Vertical dashed lines show the corresponding limits on connectivity for 50 % of spikes to belong to synfire waves (C _E,max2(n _E))

Fig. 7

The comparison of conductance-based and current-based models: (first column) conductance-based models with g _E = 0.005 and g _I ∈ [0.02,0.1]; (second column) current-based models with a _E = g _E(V _E − V _avg) and a _I ∈ [0.25, 1.5] mV; (third column) current-based models fitted to a conductance-based model with g _E = 0.005 and g _I = 0.095, for V _P ∈ [ − 88.4, − 56.6] mV; (fourth column) current-based models fitted to each of the conductance models in the first column using V _P = μ. In each column, the remaining neuronal parameters are as given in Table 1. For selected values of the varied parameter (g _I, a _I, V _P and g _I, for columns 1–4 respectively) the performance parameters μ, σ, ν _S, n _E,min and α _max are plotted against connectivity C _E up to C _E = C _E,max, the point that gives the highest embedding capacity subject to the constraints on stochastic spiking rate and stability (colored curves). The values of the varied parameter are: (first and fourth columns) g _I =0.02, 0.05, 0.07, 0.08, 0.085, 0.09, 0.095, 0.10; (second column) a _I =0.25, 0.5, 0.75, 0.875, 1., 1.125, 1.25, 1.3125, 1.375, 1.4375, 1.5 mV; and (third column) 21 values of V _P between − 88.4 mV and − 56.6 mV. In each panel the black curve is the behavior at C _E = C _E,max for the entire range of the varied parameter