Emergence of spatially periodic diffusive waves in small-world neuronal networks

It has been observed in experiment that the anatomical structure of neuronal networks in the brain possesses the feature of small-world networks. Yet how the small-world structure affects network dynamics remains to be fully clarified. Here we study the dynamics of a class of small-world networks consisting of pulse-coupled integrate-and-fire (I&F) neurons. Under stochastic Poisson drive, we find that the activity of the entire network resembles diffusive waves. To understand its underlying mechanism, we analyze the simplified regular-lattice network consisting of firing-rate-based neurons as an approximation to the original I&F small-world network. We demonstrate both analytically and numerically that, with strongly coupled connections, in the absence of noise, the activity of the firing-rate-based regular-lattice network spatially forms a static grating pattern that corresponds to the spatial distribution of the firing rate observed in the I&F small-world neuronal network. We further show that the spatial grating pattern with different phases comprise the continuous attractor of both the I&F small-world and firing-rate-based regular-lattice network dynamics. In the presence of input noise, the activity of both networks is perturbed along the continuous attractor, which gives rise to the diffusive waves. Our numerical simulations and theoretical analysis may potentially provide insights into the understanding of the generation of wave patterns observed in cortical networks.

Figure 1

The network connectivity. (a) Average path length $L\left(\beta \right)$ (red circles) and clustering coefficient $C\left(\beta \right)$ (blue squares) in the networks generated by Eq. (3) (normalized by their values in the regular-lattice limit) as function of $\beta$. We choose $N^E=N^I={10}^4$ and $p_0^E=p_0^I=0.1$. (b) The small-world connectivity profile. The neuron located at $x_i$ ($=i/N$) prefers to receive links from its local neighbors, i.e., presynaptic excitatory neurons located in the range $[x_i-p_0^E/2,x_i+p_0^E/2]$ (red) and presynaptic inhibitory neurons located in the range $[x_i-p_0^I/2,x_i+p_0^I/2]$ (blue). In addition, due to the small-world property, there also exist a few long-range connections (dashed lines).

Figure 2

Dynamics of I&F neurons in a small-world network. (a)–(c) Spike rasters of all the excitatory neurons. (a) is the case of weak connection and (b) and (c) are the cases of strong connection. (b) and (c) show the short- and long-time behavior of the network, respectively. (d) Normalized membrane potential of a sample neuron that exhibits transitions between up and down states as time evolves. (e) The spatial profile of the network firing activity in the line (0,1] (blue) fitted by the function $r\left(x\right)=r_{max}[1+\cos (\omega x+\varphi )]/2$ (red). The time-averaged firing rate of each neuron is obtained from (b). (f) Variance of phase difference $\delta \varphi$ as a function of time increment $\delta T$. The blue dots are obtained from the simulation in the I&F small-world neuronal network, and the black line is the linear fitting with slope 0.39. We set $J^E=0.001, {\nu }^E={\nu }^I=3$ kHz for (a) and $J^E=0.01, {\nu }^E={\nu }^I=6.7$ kHz for (b)–(f). Other parameters are $p_0^E=p_0^I=0.1, N^E=N^I={10}^4, f^E=f^I=0.03,$ and $J^I=2J^E$. Here the dynamics of the inhibitory population is almost the same as that of the excitatory population shown above.

Figure 3

Dynamics of the firing-rate-based neuron model. (a) Comparison of firing rates between the single firing-rate-based neuron and the I&F neuron. While receiving the inhomogeneous Poisson train with rate $14+6sin\left[0.002\pi \right(1+9t/3000\left)t\right]$ kHz, the mean firing rates of the firing-rate-based neuron and the I&F neuron are calculated over ${10}^4$ trials. The red line is for the I&F neuron, and the blue dashed line is for the firing-rate-based neuron. The strength of the Poisson input is fixed as 0.01. (b) The spatial profile of the excitatory neurons' activity for a firing-rate-based small-world network in the line $(0,1]$ in Eq. (4) for a given time without noise. The parametric values in (b) are chosen the same as those in Fig. 2. (c) Spatiotemporal profile of the excitatory neurons' activity in the same system as (b) while considering the noise of the external input. According to the Poisson external input into the I&F neuron, the variance of the external input into the firing-rate-based neuron takes the value of ${\left({\sigma }^k\right)}^2={\left(f^k\right)}^2{\nu }^k$, where $f^k$ and ${\nu }^k$ are the strength and rate of a Poisson input in the $k\mathrm{th}$ population, respectively.

Figure 4

The real part of the largest eigenvalue in different connectivity matrices. (a) The real part of the largest eigenvalue ${\lambda }_{\text{max}}^r$ of networks with the rewiring probability $\beta =0.01$ over 600 trials for $J^E=0.001$ (upper) and $J^E=0.01$ (lower). The parameters in the upper panel are chosen to be the same values as those in Fig. 2. The parameters in the lower panel are chosen to be the same values as those in Fig. 2. (b) The ratio of mean ${\lambda }_{\text{max}}^r$ of $(\mathbf{W}-\mathbf{I})$ with nonzero rewiring probability $\beta$ over trials to that of a regular-lattice network. The parameters here are set to be the same as those in Fig. 2. For each value of $\beta$, the mean largest eigenvalue is calculated over 600 trials.

Figure 5

The influence of the eigenvalue ${\lambda }_2$ on the spatial frequency of the grating pattern. (a) ${\lambda }_2\left(n\right)$ for small and large connection strengths. $J^E=0.001$ for the blue curve and $J^E=0.01$ for the red curve. Other parameters are chosen to be the same as that in Fig. 2. The black dash-dotted line is the reference line $y=0$. The eigenvalue ${\lambda }_2\approx 1.1624$ at $n^{*}=14$. (b) The Fourier coefficients of the I&F small-world network activity in Fig. 2. The global peak corresponds to the spatial frequency of the grating pattern $n^{*}=14$, which is consistent with $n^{*}$ obtained from (a).

Figure 6

Analytical solution of the spatial profile of the grating pattern. (a) A particular solution to Eq. (4) for the deterministic case in one spatial period as given in Eq. (10). In (b) and (c), the spatial profiles of the grating pattern (blue line) obtained from the simulations of the I&F small-world neuronal network can be well fitted by the particular solution given in Eq. (10) (black line). Only two spatial periods are presented here. The external input is $f^E=f^I=0.05,{\nu }^E={\nu }^I=4$ kHz for (b) and $f^E=f^I=0.001,{\nu }^E={\nu }^I=200$ kHz for (c). Other parameters are the same as in Fig. 2. (d) The portion of active neurons in the I&F small-world neuronal network as a function of the external input rate. Blue dots are from the simulations and black line is from the theoretical prediction $X_2/X$. The corresponding range of population firing rate is from 5 to 50 Hz. In order to reduce the influence of the external input noise, $f^E=f^I=0.001$. The coupling strength is $J^I=2J^E=0.066$.

Figure 7

Diffusion coefficient. (a) Comparison between $D_{\text{sim}}$ and $D_{\text{pred}}$. $D_{\text{sim}}$ is the diffusion coefficient calculated from the simulation data in the I&F small-world neuronal network and $D_{\text{pred}}$ is the diffusion coefficient predicted from Eq. (16). The values of $(D_{\text{pred}},D_{\text{sim}})$ lie near the black line $y=x$. (b) Diffusion coefficient as a function of network size $2N$. The blue dots are calculated from the simulation of the I&F small-world neuronal network, and the black line is the fit using the inversely proportional function ($\sim \frac{1}{N}$). The parameters are chosen as $J^E=0.3/\sqrt{N{p}_0}, J^I=0.6/\sqrt{N{p}_0}, N^E=N^I=N, p_0=0.1$. The external Poisson input is given to each neuron independently with strength $f^E=f^I=0.3/\sqrt{N{p}_0}$ and rate ${\nu }^E={\nu }^I=30N{p}_0$.

Figure 8

Spatial frequency $n^{*}$ as a function of parameters $p_0^E$ and $p_0^I$. The spatial frequency $n^{*}$ is analytically calculated as the value at which ${\lambda }_2\left(n\right)$ reaches its maximum in (a), approximated as $3/\left(2p_0^I\right)$ in (b), and directly computed from simulation data of the I&F small-world neuronal network in (c). The relative errors between the simulation results and the approximated value $3/\left(2p_0^I\right)$ are shown in (d). The parameters are chosen as $J^I{N}^I{p}_0^I=2J^E{N}^E{p}_0^E$ and the network size is $N^E=N^I=4000$.

Figure 9

Simulations of I&F small-world neuronal networks in higher dimensions. (a) is for a two-dimensional network of size $N^E=N^I={10}^4$ and $J^I=2J^E=0.04, f^E=f^I=0.02, {\nu }^E={\nu }^I=10$ kHz. (b) is for a three-dimensional network of size $N^E=N^I=2.7\times {10}^4, J^I=2J^E=0.07, f^E=f^I=0.03, {\nu }^E={\nu }^I=8$ kHz. Color codes active neurons in a short period of 400 ms in (a) and neurons with firing rate larger than 10 Hz (for ease of illustration) in 400 ms in (b). Other parameters are $p_0^E=p_0^I=0.08.$