Microscopic instability in recurrent neural networks

In a manner similar to the molecular chaos that underlies the stable thermodynamics of gases, a neuronal system may exhibit microscopic instability in individual neuronal dynamics while a macroscopic order of the entire population possibly remains stable. In this study, we analyze the microscopic stability of a network of neurons whose macroscopic activity obeys stable dynamics, expressing either monostable, bistable, or periodic state. We reveal that the network exhibits a variety of dynamical states for microscopic instability residing in a given stable macroscopic dynamics. The presence of a variety of dynamical states in such a simple random network implies more abundant microscopic fluctuations in real neural networks which consist of more complex and hierarchically structured interactions.

Figure 1

A recurrent network of neurons. $w_{ij}$ represents a synaptic connection from the $j\text{th}$ neuron to the $i\text{th}$ neuron. Neurons update their states according to Eq. (2).

Figure 2

Phase diagram of the macrodynamics in a plane of $\overline{w}$ and $h$: P is the periodic state, Sm the monostable state, and Sb the bistable state.

Figure 3

Examples of the three types of dynamics of the macroscopic activity: (a) monostable $(\overline{w}=1,h=0.5)$, (b) bistable $(\overline{w}=2,h=0)$, and (c) periodic state $(\overline{w}=-2,h=0)$.

Figure 4

Microscopic stability of the macroscopically stable states ($N=1000$). A: Both macroscopically stable states are microscopically stable. B and ${\text{B}}^{\prime }$: One of two macroscopically stable states is microscopically unstable. C: Both macroscopically stable states are microscopically unstable. D and ${\text{D}}^{\prime }$: The macroscopically monostable state is microscopically stable. E: The macroscopically monostable state is microscopically unstable.

Figure 5

Microscopic stability of macroscopically periodic states ($N=1000$). F: Both macroscopic states are microscopically unstable. G and ${\text{G}}^{\prime }$: One of two macroscopic states is microscopically unstable. H: Both macroscopic states are microscopically stable.

Figure 6

Boundaries of microscopic stability. The boundaries for $N=1000,10000$, and $100000$ are depicted in red (solid), brown (dashed), and purple (dotted), respectively. Regions A–H correspond to the respective regions given in Figs. 4 and 5.

Figure 7

Microscopic instability obtained by numerical simulation of the networks of $N=1000$. A set of parameters $(\overline{w},h)$ with which the network exhibits instability is depicted as a dot. Initial states of neurons were chosen at random, $s_i=\pm 1$.

Figure 8

Microscopic instability obtained from the set of initial states of $\{s_i=+1\}$ ($N=1000$).

Figure 9

Distance maps $\phi \left(d\right)$ for various $h$ ($\overline{w}=0$).

Figure 10

Change in the distance map $\phi \left(d\right)$ across the microscopic stability line ($N=1000$): (a) three kinds of distance maps and (b) model parameters for the three distance maps.

Figure 11

Period of microscopic states and the number of neurons.

Figure 12

Dependence of the exponent $\gamma$ on the model parameter $h$ ($\overline{w}=0$, $N=30$). ${\gamma }_B$ is the exponent estimated by the Boolean random map given a typical number of microscopic states.