Approximate-master-equation approach for the Kinouchi-Copelli neural model on networks

In this work, we use the approximate-master-equation approach to study the dynamics of the Kinouchi-Copelli neural model on various networks. By categorizing each neuron in terms of its state and also the states of its neighbors, we are able to uncover how the coupled system evolves with respective to time by directly solving a set of ordinary differential equations. In particular, we can easily calculate the statistical properties of the time evolution of the network instantaneous response, the network response curve, the dynamic range, and the critical point in the framework of the approximate-master-equation approach. The possible usage of the proposed theoretical approach to other spreading phenomena is briefly discussed.

Figure 1

Schematic illustration of the approximate-master-equation approach for the Kinouchi-Copelli neural model. The solid arrows represent the changes in the states of the neurons' neighbors (subscript transformation), and the dashed arrows represent the changes in the state of the focal neuron $X$ itself ($X\in \{R,S,T\}$, state transformation). $X_{\alpha ,\beta ,\gamma }$ is an intermediate state, the expressions $Q_{\alpha +\beta -l,\gamma ,l}$ ($Q\in \{G,H,O\},l\in \{0,1,2,...,\alpha \}$) denote the probabilities of the subscript transformation, and the expressions above the dashed arrows indicate the probabilities of the change of different $X$. See more details in the corresponding text.

Figure 2

Time series of the network instantaneous response ${\rho }_t$ on the ER random networks with $K=6$ and without an external stimulus (i.e., $r=0$). (a) In the subcritical regime with $p=0.14$, (b) at the critical point with $p=1/6$, (c) in the supercritical regime with $p=0.2$, and (d) in the supercritical regime with $p=0.8$. Results from the AMEA are depicted by the thick solid lines (for just one realization), and those from stochastic simulations are represented by the thin solid lines (for 100 independent realizations).

Figure 3

The network instantaneous response ${\rho }_t$ as a function of time $t$ on (a) the ER random network with $K=6$, (b) the RRG with $K=6$, and (c) the uncorrelated random SF network with $\gamma =2.5$ and without external stimulus ($r=0$). Results from the AMEA are given by the thin solid lines, and the least-squares estimation of the closest power-law form for the critical state with $100\le t\le 1000\mathrm{ms}$ is displayed by thick solid lines.

Figure 4

Response curves $F$ of the Kinouchi-Copelli neural model on (a) ER random networks with $K=6$ (from $p=0$ to $1/3$ in intervals of $1/30$), (c) RRGs with $K=6$ (from $p=0$ to 0.4 in intervals of 0.04), and (e) uncorrelated random SF networks with $\gamma =2.5$ (from $p=0$ to $0.22874$ in intervals of $0.022874$). Results from stochastic simulations on uncorrelated random SF networks are averaged over 100 independent realizations. Dynamic range $\mathrm{\Delta }$ vs the branching probability $p$ on (b) ER random networks with $K=6$, (d) RRGs with $K=6$, and (f) uncorrelated random SF networks with $\gamma =2.5$. The olive squares in (f) represent the results for $p=0.12$–0.13 (in intervals of 0.001) with the additional stochastic simulations. The dashed and solid arrows represent the critical points obtained by stochastic simulations and the AMEA, respectively.

Figure 5

The exponent $h$, which is obtained by doing the least-squares fitting of the response curves $F$ with ${10}^{-5}\le r\le {10}^{-3}{\mathrm{ms}}^{-1}$ according to the relationship $F\sim r^h$, as a function of the branching probability $p$ on (a) ER random networks, (b) RRGs, and (c) uncorrelated random SF networks. Parameters are the same as those in Fig. 4. Clearly, the exponent $h$ is close to 1 for the uncoupled networks ($p=0$), close to 1/2 at the critical point ($p=p_c$), and decreases to 0 in the supercritical regime ($p=1$).

Figure 6

Minimum response $F_0$ as a function of the branching probability $p$ of the Kinouchi-Copelli neural model on (a) ER random networks, (b) RRGs, and (c) uncorrelated random SF networks.

Figure 7

The threshold $p_c$ vs the average degree $K$ on (a) ER random networks and (b) RRGs. The black squares are simulation results, the red circles are the results from the AMEA, the dashed lines are the linear fitting of the AMEA results in the log-log scale, and the solid line is the solution of Eq. (13). Other parameter values are the same as those in Fig. 6.

Figure 8

Schematic illustration of the transitions of the dynamics of the asynchronous update model. There is only one neuron updating its state at each time step. The dashed arrows represent the change of neuronal states, while the solid arrows and the dotted arrows represent the changes induced by the state transformation of the neuron's neighbors. Expressions over the lines are the probabilities of transformations.

Figure 9

(a) Response curves $F$ as a function of $r$ of the asynchronous update model on ER random networks with $N={10}^4$, $K=6$ (from $p=0$ to $0.36380$ in intervals of $0.03638$). (b) Dynamic range $\mathrm{\Delta }$ vs the branching probability $p$ on ER random networks. (c) Minimum response $F_0$ on the ER random networks. Results from stochastic simulations are averaged over 10 independent realizations.