Optimal spike-based communication in excitable networks with strong-sparse and weak-dense links

Abstract

The connectivity of complex networks and functional implications has been attracting much interest in many physical, biological and social systems. However, the significance of the weight distributions of network links remains largely unknown except for uniformly- or Gaussian-weighted links. Here, we show analytically and numerically, that recurrent neural networks can robustly generate internal noise optimal for spike transmission between neurons with the help of a long-tailed distribution in the weights of recurrent connections. The structure of spontaneous activity in such networks involves weak-dense connections that redistribute excitatory activity over the network as noise sources to optimally enhance the responses of individual neurons to input at sparse-strong connections, thus opening multiple signal transmission pathways. Electrophysiological experiments confirm the importance of a highly broad connectivity spectrum supported by the model. Our results identify a simple network mechanism for internal noise generation by highly inhomogeneous connection strengths supporting both stability and optimal communication.

Figure 1. Maximizing the fidelity of spike transmission with long-tailed sparse connectivity.

(a) Each excitatory neuron has a lognormal amplitude distribution of EPSPs. The resultant mean and variance of the model are 0.89 [mV] and 1.1² [mV²], respectively, whereas those shown in a previous experiment [1] were 0.77 [mV] and 0.9² [mV²]. Inset is a normal plot of the same distribution. (b) Schematic illustration of the neuron model with strong-sparse and weak-dense synaptic inputs. Colors (red, green and blue) indicate inputs to the top three strongest weights. (c) C.C.s between the output spike train and input spike trains at the 1st (red), 2nd (green) and 3rd (blue) strongest synapses on a neuron are plotted against the mean membrane potential and the corresponding input firing rate at each synapse. The dashed line and shaded area show the mean and SD of the membrane potential distribution of excitatory neurons shown in Fig. 2f for the SSWD network. Vertical bars represent SEM over different realizations of random input. The dashed line indicates an analytical curve for the strongest synapse of the long-tailed distribution, while the dot-dashed line is the C.C.s for the strongest synapse when EPSP amplitudes obey Gaussian distribution. (d) Similar C.C.s obtained by dynamic clamp recordings from a cortical neuron. The color code and vertical bars are the same as in C. (e) The trial-averaged C.C.s for the strongest synapses on n = 14 neurons.

Figure 2. Spontaneous noise in the SSWD recurrent network.

The network receives neither external input nor background noise, and hence activity is spontaneous. (a) Upper, Spike raster of excitatory (red) and inhibitory (blue) neurons in the noisy spontaneous firing state. Lower, The population firing rates of excitatory (red) and inhibitory (blue) neurons. (b) Firing rate distributions of excitatory (red) and inhibitory (blue) neurons can be fitted by lognormal distributions (black lines). Mean firing rates are 1.6 and 14 [Hz] for excitatory and inhibitory neurons respectively. (c) CVs of inter-spike intervals are distributed around unity in excitatory (red) and inhibitory (blue) neurons. (d) Time courses of the membrane potentials of excitatory (red) and inhibitory (blue) neurons exhibit large amplitude fluctuations. (e) Scatter plot of the instantaneous population activities of excitatory and inhibitory neurons. The solid line represents linear regression. (f) Distribution functions of the fluctuating membrane potentials show the depolarized states of excitatory (red) and inhibitory (blue) neurons. (g) The mean (solid) and standard deviation (dashed) of the membrane potential fluctuations of an excitatory neuron when all EPSPs smaller than the minimum value given in the abscissa are eliminated. Here, we remove a portion of excitatory synapses on a neuron from the weakest ones.

Figure 3. Spike information routing in the SSWD recurrent network model.

(a) A schematic illustration of the SSWD recurrent network. Thick lines stand for strong-sparse connections and thinner lines for weak-dense connections. In reality, the strength of connections is continuous obeying a long-tailed distribution. (b) Examples of spike sequences routed in the network are shown by different colors. Insets magnify the raster plot. (c) Examples of branching (left) and converging (right) pathways formed by the strong synapses. Numbers refer to neurons, and colors to the corresponding pathways in (b). (d) Cross-correlograms are averaged over strongly connected neuron pairs (EPSP >8mV). (e) Repeated external stimuli (arrows) evoke simultaneous spike propagations in two pathways. (f) Linear relationship between the number of input spikes and that of output ones in a pathway. The dashed line and vertical bars represent linear regression and SD over trials, respectively.