Synchronization and long-time memory in neural networks with inhibitory hubs and synaptic plasticity

We investigate the dynamical role of inhibitory and highly connected nodes (hub) in synchronization and input processing of leaky-integrate-and-fire neural networks with short term synaptic plasticity. We take advantage of a heterogeneous mean-field approximation to encode the role of network structure and we tune the fraction of inhibitory neurons $f_I$ and their connectivity level to investigate the cooperation between hub features and inhibition. We show that, depending on $f_I$, highly connected inhibitory nodes strongly drive the synchronization properties of the overall network through dynamical transitions from synchronous to asynchronous regimes. Furthermore, a metastable regime with long memory of external inputs emerges for a specific fraction of hub inhibitory neurons, underlining the role of inhibition and connectivity also for input processing in neural networks.

Figure 1

${\langle \text{ISI}\rangle }_k$ as a function of the degree $k$ for a network of $N=5000$ neurons, which evolves according Eqs. (1, 2, 3, 4, 5, 6, 7). In the inset we show a raster plot of the corresponding dynamics, where nodes are ordered along the vertical axis according to their degree $k$; a dot means that neuron $i$ has fired at time $t$. Here and in all other plots time is dimensionless. The inhibitory fraction $f_I$ is 0.1. Blue (green) data refer to the excitatory (inhibitory) nodes. The excitatory average degree $\langle k_E\rangle$ is 100, the inhibitory one $\langle k_I\rangle$ is 350 and the standard deviation $\sigma$ of both Gaussian p.d.f.s $P_E\left(k\right)$ and $P_I\left(k\right)$ is 10.

Figure 2

(a) $\langle {\text{ISI}}_k\rangle$ as a function of the degree $k$ for networks with different inhibitory fractions $f_I$ (see the legend). $\mathrm{\Delta }$ is fixed at 250 and the number $M$ of connectivity classes is 500. In the inset we show a zoom of the excitatory plateau. In panel (b) we display snapshots of the raster plots for different inhibitory fractions, to represent the three main dynamical regimes: partial synchronization for $f_I=0.1$, total synchronization for $f_I=0.22$, and an asynchronous state for $f_I=0.29$. A dot in raster plots of HMF simulations represents the spike event of connectivity class $i$ at time $t$ and classes are ordered along the vertical axis according to their degree $k$. The degree classes below (above) the black line are excitatory (inhibitory). These plots refer to different dynamical regimes, however a time unit in the abscissa axis corresponds to about one global oscillation of the network (after this time interval all neurons have fired at least once).

Figure 3

Network synchronization from the measure of $1-R$ (where $R$ is the Kuramoto parameter) as a function of $f_I$ for different values of $\mathrm{\Delta }$ (see the legend), in semilogarithmic scale. In all networks $M$ is equal to 500. The closer to 0 the parameter $1-R$ is, the more synchronous the network. The grey-shaded area represents the interval of inhibitory fractions, which has been really observed in some brain tissues. Without hub nodes ($\mathrm{\Delta }=0$, black data), the network reaches a high synchronization level out of this region, differently from $160\lesssim \mathrm{\Delta }\lesssim 500$. Therefore, if we want to build synchronous networks with inhibitory fractions close to real values, the inhibitory neurons should be hubs, with an average degree $3–6$ times greater than the excitatory average degree.

Figure 4

Relative weights of the excitatory and the inhibitory component in driving the networks dynamics with $M=500$ connectivity classes, from the measure of $W_E$ and $W_I$ for different values of $\mathrm{\Delta }$ (see the legend) and $f_I$. The balance value $f_I^B$ of the inhibitory fraction for which $W_E=W_I=0$ is 0.5 for $\mathrm{\Delta }=0$, 0.28 for $\mathrm{\Delta }=160$, 0.22 for $\mathrm{\Delta }=250$, 0.143 for $\mathrm{\Delta }=500$ and 0.083 for $\mathrm{\Delta }=1000$.

Figure 5

Raster plots of a metastable partially synchronous state for the network with $M=500$, $\mathrm{\Delta }=250$, and $f_I=0.214$, starting from random initial conditions. These plots are two snapshots, taken at different times during the same dynamical evolution. The degree classes below (above) the black line are excitatory (inhibitory).

Figure 6

Raster plot of a network with $M=500$ connectivity classes, $\mathrm{\Delta }=250$ and $f_I=0.1$. At a given time (see the red arrow), we apply a synchronizing external stimulus to $30\%$ of excitatory neurons, randomly chosen. After about ten global oscillations, the stimulated nodes recover the initial dynamics. The degree classes below (above) the black line are excitatory (inhibitory). Light (dark) blue dots represent the spike events of stimulated (not stimulated) nodes.

Figure 7

Transient dynamics at the stimulus offset: return to the original synchronization level after the application of an external input. We plot the parameter $1-R\left(t\right)$ of the stimulated nodes as a function of time, for networks with $M=500$, $\mathrm{\Delta }=250$, and different values of $f_I$ (see the legend). The external synchronous stimulus is always applied at $t=0$ to $30\%$ of excitatory neurons, randomly chosen.

Figure 8

Response of the network to external stimuli applied at different times and with different intensity. In both figures, we plot the parameter $1-R\left(t\right)$ of the stimulated nodes as a function of time and we consider networks with $M=500$ and $\mathrm{\Delta }=250$. Green (red) shades data refer to $f_I=0.1\left(0.214\right)$. Upper panel: for each network, we choose four different times to switch on a stimulus with the same intensity, i.e., on $30\%$ of excitatory neurons, randomly chosen. All data are shifted so that the onset of the perturbation is at $t=0$. Lower panel: we compare the transient dynamics for stimuli applied on $10\%$, $20\%$, and $30\%$ of the excitatory neurons at $t=0$.

Figure 9

Comparison of the dynamics of a network with $f_I=0.1$, simulated through both the finite size and the HMF approach. In the upper figures red data are produced evolving the finite size equations with $N=6000$ neurons, and blue data the HMF ones with $M=500$ connectivity classes. The p.d.f. is characterized in both cases by $\langle k_E\rangle =100$, $\mathrm{\Delta }=200$, and $\sigma =10$. In the lower figures we show how the HMF approximation improves as we approach the limit of $k_i\to \infty$. Again, green (blue) data evolve according to the finite size (HMF) approach, but for the finite size network we have quadrupled the degrees (new p.d.f. parameters: $\langle k_E\rangle =400$, $\mathrm{\Delta }=800$, $\sigma =40$). Thanks to this higher connectivity, the finite size dynamics better converges to the same HMF network. In panels (a) we compare the raster plots. As in the HMF figures $M$ is equal to 500, the HMF indices have been opportunely rescaled for a better overlap with the red and green data. Data below (above) the black line refer to excitatory (inhibitory) nodes. (b) Field $Y_E\left(t\right)$ as a function of time. The comparison of $Y_I\left(t\right)$ gives similar results. (c) $\langle {\text{ISI}}_k\rangle$ as a function of the degree $k$. In the lower panel, we divided the degrees of green data by 4, to overlap finite size data with HMF points. The inset is a zoom of the $\langle {\text{ISI}}_k\rangle$ only for the excitatory neurons and for all three sets of data. (d) Attractors $Y_E\left(t\right)$ vs $Z_E\left(t\right)$.

Figure 10

Stability of the synchronous state in finite size simulations. In the upper panel we compare the time evolution of the parameter $1-R\left(t\right)$ (in logarithmic scale) in three different networks with $N=6000$ neurons and $f_I=0.25$. The p.d.f. of the first network (blue data) is characterized by $\mathrm{\Delta }=200$, $\langle k_E\rangle =100$, $\sigma =10$; the p.d.f. of the second network is doubled (dark green data, $\mathrm{\Delta }=400$, $\langle k_E\rangle =200$, $\sigma =20$); the p.d.f. of the last network is quadrupled (light green data, $\mathrm{\Delta }=800$, $\langle k_E\rangle =400$, $\sigma =40$). The external stimulus is applied at time $t=0$ to all neurons. At the beginning $1-R\left(t\right)$ is almost null, because the system is totally synchronous, but then due to the finite size fluctuations it returns to 1, that is the network evolves towards its stable asynchronous state. In correspondence of the black arrows, we show in the three lower figures the microscopic dynamics of the networks through their raster plots. The indices below (above) the black line refer to excitatory (inhibitory) neurons.

Figure 11

Dynamics of finite size networks after the application of an external stimulus at time $t=0–30\%$ of excitatory neurons, randomly chosen, with $N=5000$ neurons, $\mathrm{\Delta }=250$, $\langle k_E\rangle =100$, $\sigma =10$ and different inhibitory fractions $f_I$ (see the legend). For each network we plot in the upper panel (in semilogarithmic scale) the temporal average values $\langle \mathrm{\Delta }{Y}_{EE}\left(t\right)\rangle$ as a function of time, starting from $t=0$. In correspondence of the black arrows, in the lower panels we show snapshots of the raster plot taken at different times for the network with $f_I=0.1$ [(a) and (b)] and with $f_I=0.214$ [(c) and (d)]. The spike events of stimulated (nonstimulated) nodes are represented by light (dark) blue dots, while the orange and red lines are the signals defined in Eq. (19). To better overlap signals with raster plots, we shifted them adding a positive constant: this is why they are not null before the stimulus. After the stimulus, signals turn on sharply and then decay. The indices below (above) the black line refer to excitatory (inhibitory) neurons.