Heterogeneous mean field for neural networks with short-term plasticity

We report about the main dynamical features of a model of leaky integrate-and-fire excitatory neurons with short-term plasticity defined on random massive networks. We investigate the dynamics by use of a heterogeneous mean-field formulation of the model that is able to reproduce dynamical phases characterized by the presence of quasisynchronous events. This formulation allows one to solve also the inverse problem of reconstructing the in-degree distribution for different network topologies from the knowledge of the global activity field. We study the robustness of this inversion procedure by providing numerical evidence that the in-degree distribution can be recovered also in the presence of noise and disorder in the external currents. Finally, we discuss the validity of the heterogeneous mean-field approach for sparse networks with a sufficiently large average in-degree.

Figure 1

Raster plot of a randomly diluted network containing 500 neurons, ordered along the vertical axis according to their in-degree. The distribution $P\left(\tilde{k}\right)$ is a Gaussian with $\langle \tilde{k}\rangle =0.7$, standard deviation ${\sigma }_{\tilde{k}}=0.077$. A black dot in the raster plot indicates that neuron $s$ has fired at time $t$. The maroon (dark gray) crosses are the global field $Y\left(t\right)$ and the green (light gray) continuous curve is its analytic fit by the function $Y_f\left(t\right)=A{e}^{-\frac{t}{{\tau }_1}}+B(e^{\frac{t}{{\tau }_2}}-1)$ that repeats over each period of $Y\left(t\right)$; the parameter values are $A=2\times {10}^{-2}$, $B=3.56\times {10}^{-6}$, ${\tau }_1=0.268$, and ${\tau }_2=0.141$. Notice that the amplitude of both $Y\left(t\right)$ and $Y_f\left(t\right)$ has been suitably rescaled to be appreciated on the same scale of the Raster plot.

Figure 2

The effect of sampling the probability distribution $P\left(\tilde{k}\right)$ with $M$ classes of neurons in the HMF dynamics. Finite-size effects are controlled by plotting the distance between the activity fields obtained for two sampling values $M$ and $M/2$, $d_M=d(Y_M\left(t\right),Y_{M/2}\left(t\right))$ (defined in the text), vs $M$. The red dashed line is the power law $1/\sqrt{M}$. Data are obtained for a Gaussian distribution $P\left(\tilde{k}\right)$, with $\langle \tilde{k}\rangle =0.7$ and ${\sigma }_{\tilde{k}}=0.077$.

Figure 3

The return map $R_{\tilde{k}}$ of the rescaled variable ${\tau }_{\tilde{k}}/T$ [see Eq. (16)] for different values of $\tilde{k}$, corresponding to different line styles with different colors (see the legend in the inset: the black line is the bisector of the square).

Figure 4

The maximum Lyapunov exponent ${\lambda }_{\max }$ as a function of the sampling parameter $M$: ${\lambda }_{\max }$ has been averaged also over 10 different realizations of the network (the error bars refer to the maximum deviation from the average). The dashed red line is the power law $M^{-\gamma }$, with $\gamma =0.55$.

Figure 5

The time average of the interspike interval $\overline{T_{\tilde{k}}^{\mathrm{isi}}}$ vs $\tilde{k}$ for the probability distribution $P\left(\tilde{k}\right)$ defined in Eq. (20), with $\Delta =|p_2-p_1|=0.4$ and ${\sigma }_s=0.03$. We have obtained the global field $Y\left(t\right)$ simulating the HMF dynamics with a discretization with $M=300$ classes of neurons. We have then used $Y\left(t\right)$ to calculate $T^{\mathrm{isi}}$ of neurons evolving Eq. (10). In the inset we show the raster plot of the dynamics: as in Fig. 1, neurons are ordered along the vertical axis according to their in-degree.

Figure 6

The frequency spectra of the global activity field $Y\left(t\right)$ for different in-degree probability distributions. The black spectrum with stars has been obtained for the HMF dynamics with $M=350$, generated by the power-law probability distribution $P\left(\tilde{k}\right)\sim {\tilde{k}}^{-4.9}$ [see Eq. (19)], with ${\tilde{k}}_m=0.1$: In this case there is a unique family of locked neurons generating a periodic global activity field $Y\left(t\right)$. The red continuous line spectrum has been obtained for a random network of $N=300$ neurons generated by the double Gaussian distribution [see Eq. (20)] described in Figs. 6 and 7: In this case two families of locked neurons are present while, as reported in the inset, $Y\left(t\right)$ exhibits a quasiperiodic evolution.

Figure 7

The global activity field $Y\left(t\right)$ of the HMF dynamics, sampled by $M=4525$ classes of neurons, for a Gaussian probability distribution $P\left(\tilde{k}\right)$, with $\langle \tilde{k}\rangle =0.7$ and ${\sigma }_{\tilde{k}}=0.0455$. Lines of different colors and style correspond to different values of the noise amplitude, $\delta$, added to the external currents $a_{\tilde{k}}\left(t\right)$. Notice that increasing $\delta$ decreases the peak of the curve. In particular, $\delta =0$ (black continuous line), $\delta =0.1$ (red dots), $\delta =0.15$ (green segments), $\delta =0.2$ (blue dots and segments), and $\delta =0.3$ (gray continuous line).

Figure 8

Solution of the inverse problem by the HMF equations in the presence of noise added to the external currents. We consider the same setup of Fig. 9 and we compare, for different values of the noise amplitude $\delta$, the reconstructed probability distribution $P\left(\tilde{k}\right)$ (red circles) with the original Gaussian distribution (black line): the upper-left panel corresponds to the noiseless case ($\delta =0$), while the upper-right, the lower-left, and and the lower-right panels correspond to $\delta =0.1,0.2,0.3$, respectively.

Figure 9

Solution of the inverse problem by the HMF equations in the presence of noise added to the activity field. We consider the same setup of Fig. 9, where now $a=1$ and $Y_{\delta }\left(t\right)=[1-\eta \left(t\right)]Y\left(t\right)$ (the random variable $\eta \left(t\right)$ is extracted from a uniform probability distribution in the interval $[-\delta /2,\delta /2]$). We compare, for different values of the noise amplitude $\delta$, the reconstructed probability distribution $P\left(\tilde{k}\right)$ (red circles) with the original Gaussian distribution (black line): the upper-left, upper-right, lower-left, and lower-right panels correspond to $\delta =0.1,0.4,0.8,1.2$, respectively.

Figure 10

Comparison of the global synaptic activity field $Y\left(t\right)$ from sparse random networks with the same quantity generated by the corresponding HMF dynamics. We have considered sparse random networks with $N={10}^4$ neurons. In the upper panel we consider a Gaussian probability distribution $P\left(k\right)$ with different averages $\langle k\rangle$ and variances ${\sigma }_k$, such that ${\sigma }_k/\langle k\rangle =0.06$: $\langle k\rangle =10,20,60,100$ correspond to the gray continuous curve, orange dots and segments, and red segments and blue dots, respectively. The black continuous line represents $Y\left(t\right)$ from the HMF dynamics ($M={10}^3$), where $\widehat{P}\left(\widehat{k}\right)$ is a Gaussian probability distribution with $\langle \widehat{k}\rangle =1$ and ${\sigma }_{\widehat{k}}={\sigma }_k/\langle k\rangle =0.06$. In the lower panel we consider the scale-free case with fixed power exponent $\alpha$ and different $k_m$: $k_m=10,30,70$ correspond to the gray continuous line, red segments, and blue dots, respectively. The black continuous line represents $Y\left(t\right)$ from the HMF dynamics ($M={10}^3$), where $\widehat{P}\left(\widehat{k}\right)=(\alpha -1){\widehat{k}}^{-\alpha }$ with cutoff ${\widehat{k}}_m=1$. Notice that in both cases decreasing $\langle k\rangle$ decreases the peak of the curve with respect to the HMF curve peak.