Reaction-diffusion-like formalism for plastic neural networks reveals dissipative solitons at criticality

Self-organized structures in networks with spike-timing dependent synaptic plasticity (STDP) are likely to play a central role for information processing in the brain. In the present study we derive a reaction-diffusion-like formalism for plastic feed-forward networks of nonlinear rate-based model neurons with a correlation sensitive learning rule inspired by and being qualitatively similar to STDP. After obtaining equations that describe the change of the spatial shape of the signal from layer to layer, we derive a criterion for the nonlinearity necessary to obtain stable dynamics for arbitrary input. We classify the possible scenarios of signal evolution and find that close to the transition to the unstable regime metastable solutions appear. The form of these dissipative solitons is determined analytically and the evolution and interaction of several such coexistent objects is investigated.

Figure 1

Two adjacent layers in the considered feed-forward architecture. Synapses between layers obey the plasticity rule (1).

Figure 2

Stable solution in the critical regime. (a) Explicit $\check{f}\left(r\right)$ (black) and implicit $K(w_0+\frac{\gamma }{\alpha }{r}^2)r$ nonlinearity (gray). (b) Their difference, $q\left(r\right)$. (c) $Q\left(r\right)=\int q\left(r\right)dr$, a minimum of $Q$ of 0 enables a stable non-trivial solution $r\left(x\right)$ shown in (d)—theoretical prediction of the stable solution with dashed line and simulation results with solid line. Simulation parameters: $N=800, M=400, K=41, w_0=0.99/K, \gamma =0.99/K, \theta =0.6, \beta =3.6, A=1.0754$.

Figure 3

Single bump evolution in subcritical regime. (a) $Q\left(r\right)$ near its minimum close to but bigger than 0. (b) Rate profiles $r\left(x\right)$ in different layers showing the reduction of the plateau's width for a long-living rate profile leading to linear decrease with the layer's number of $\int r\left(x\right)dx$. (c) The theoretical expectation (12) and the simulated result for $\int r\left(x\right)dx$. Parameters as for Fig. 2 apart from $A=1.0745$.

Figure 4

Coexisting bumps in sucritical regime. (a) Theoretical prediction of the change of minimal activity $r_{\min }$ between bumps after one layer. (b) Propagation of two bumps uniting to one. (c) Two bumps remain disjointed, because their initial distance exceeds the maximal distance for unification. (d) Rate profiles in three layers. The activity of the bumps is shifted upwards in proportion to the layer number for clearer visual display. The evolution over layers illustrates the construction of an “association tree”: first the left and the middle bumps are joint, and later the right one. Parameters as for Fig. 3 apart from $\gamma =0, w_0=1.4/K, \beta =3.63$ for (a)–(c).