Stochastic mean-field formulation of the dynamics of diluted neural networks

We consider pulse-coupled leaky integrate-and-fire neural networks with randomly distributed synaptic couplings. This random dilution induces fluctuations in the evolution of the macroscopic variables and deterministic chaos at the microscopic level. Our main aim is to mimic the effect of the dilution as a noise source acting on the dynamics of a globally coupled nonchaotic system. Indeed, the evolution of a diluted neural network can be well approximated as a fully pulse-coupled network, where each neuron is driven by a mean synaptic current plus additive noise. These terms represent the average and the fluctuations of the synaptic currents acting on the single neurons in the diluted system. The main microscopic and macroscopic dynamical features can be retrieved with this stochastic approximation. Furthermore, the microscopic stability of the diluted network can be also reproduced, as demonstrated from the almost coincidence of the measured Lyapunov exponents in the deterministic and stochastic cases for an ample range of system sizes. Our results strongly suggest that the fluctuations in the synaptic currents are responsible for the emergence of chaos in this class of pulse-coupled networks.

Figure 1

Characterization of the field fluctuations for a DD network. (a) Macroscopic attractor reported in the $(\overline{E},\overline{P})$ plane (black dots); the colormap (superimposed on the attractor) quantifies the time-averaged values of the fluctuations $\langle {\sigma }_E(\overline{E},\overline{P})\rangle$. These are estimated over a $100\times 100$ grid with resolution $\Delta \overline{E}=0.06$ and $\Delta \overline{P}=0.8$. (b) PDFs $F\left(d_i\right)$ of the deviation from the average field $d_i$ estimated in three different points along the attractor. These points are indicated in panel (a) with the same symbol and color code. (c) Fluctuations of the fields $\langle {\sigma }_E\rangle$ averaged both in time and all along the whole attractor as a function of the system size $N_D$ (filled circles). The dashed line indicates a power-law fitting to the data, namely $\langle {\sigma }_E\rangle \propto N_D^{-0.6}$. (d) Autocorrelation function $C_E\left(\tau \right)$ of the fluctuations of the fields $d_i$. For all the reported data the parameters are fixed to $g=0.5$, $\alpha =9$, $a=1.05$, and $K=0.2N_D$. The system size is set to $N_D=500$, apart in panel (c). The reported quantities have been evaluated over ${10}^5–{10}^6$ spikes, after discarding an initial transient of ${10}^6$ spikes.

Figure 2

Reconstruction of the macroscopic dynamics of the DD system (black filled circles) in terms of white (blue empty triangles) and colored noise (red empty squares). [(a)–(d)] Histograms of the macroscopic fields $E$ and $P$; [(e) and (f)] macroscopic attractors; [(g)–(h)] time traces of the field $E$; (i) average period of the field $\langle T_E\rangle$ as a function of the system size $N_D$ of the DD system. Panels (a), (b), (e), and (g) refer to $N_D=500$, while panels (c), (d), (f), and (h) refer to $N_D=5000$. In (g) and (h) the time traces have been shifted in order to ensure for the coincidence of the time of occurrence of the first maximum in each trace; moreover, the colored (white) noise has been identified with a dashed (dashed-dotted) line for the sake of visualization in (g), (h), and (i). The periods reported in (i) have been obtained by measuring the time lapse between two consecutive maxima, the number of samples used for the calculation of $T_E$ is 5000 data points, and the (green) dot-dashed line is the field period in a corresponding FC deterministic network equal to 1.98. Reconstructed dynamics have been obtained with $N_S=100$ with an integration step $h=5\times {10}^{-6}$. Other parameters are as in Fig. 1.

Figure 3

Reconstruction of the microscopic dynamics of the DD system (black filled circles) in terms of white (blue empty triangles) and colored noise (red empty squares). [(a) and (c)] PDF $F\left(\mathrm{ISI}\right)$ of the ISIs; [(b) and (d)] ISIs return maps. Panels (a) and (b) refer to $N_D=500$, while panels (c) and (d) to $N_D=5000$. (e) Average ISI as a function of the DD system size $N_D$, the thick (green) dot-dashed line refers to the asymptotic value of the average ISI in the corresponding FC deterministic set up, equal to 1.96 time units. Also the colored (white) noise has been identified with a dashed (dashed-dotted) line for the sake of visualization in (e). The reconstructed dynamics are obtained with $N_S=100$ by employing an integration step $h=5\times {10}^{-6}$. The other parameters are as in Fig. 1.

Figure 4

Reconstruction of the microscopic and macroscopic dynamics for a representative size of the DD system in the limit of large dilution. From left to right, reconstruction of the distributions of the instantaneous fields $E$, $P$ and of the single-neuron ISIs. The considered DD network (black filled circles) has a size $N_D=10000$ and an in-degree $K=100$, while the reconstructed dynamics, both for the white noise (blue empty triangles) and the colored noise (red empty squares), are obtained with $N_S=100$, integrated with a step $h=5\times {10}^{-6}$. Other parameters are as in Fig. 1.

Figure 5

Maximal Lyapunov exponent ${\lambda }_M$ as a function of the system size $N_D$ for the DD case (black filled circles and solid line) and the corresponding stochastic reconstructions evaluated with the MDPH method (red squares) and the OPT approach (blue triangles). The stochastic results are reported for white (empty symbols and dotted lines) and Ornstein-Uhlenbeck (filled symbols and dashed lines) noise. In panel (a) we report the results for massively connected networks, where $K=0.2\times N_D$ and in (b) for sparse networks where $K=100$ independent from the value of $N_D$. The inset of panel (a) shows in double logarithmic scale ${\lambda }_M$ versus $N_D$ for the massively connected networks. Reconstructed dynamics obtained with $N_S=100$ and an integration step of $h=5\times {10}^{-6}$. In all cases, the system is relaxed during a transient of ${10}^6$ spikes and the Lyapunov exponents are calculated by integrating the tangent space for a period corresponding to ${10}^7-{10}^8$ spikes. Other parameters are as in Fig. 1.