Memory Network with Dynamic Synapses

(1)

$\displaystyle\begin{array}{rcl} x_{i}(t + 1)& =& x_{i}(t) + \frac{1 - x_{i}(t)} {\tau _{R}} - s_{i}(t)x_{i}(t)u_{i}(t),{}\end{array}$

(2)

$\displaystyle\begin{array}{rcl} u_{i}(t + 1)& =& u_{i}(t) + \frac{U_{se} - u_{i}(t)} {\tau _{F}} + U_{se}(1 - u_{i}(t))s_{i}(t),{}\end{array}$

(3)

where $h_{i}(t) =\sum _{ j\neq i}^{N}J_{ij}[2s_{j}(t)x_{j}(t)u_{j}(t)/U_{se} - 1]$ represents the total input to the ith neuron and $1/\beta = T$ represents the noise intensity. The quantity J _ijrepresents the absolute strength of the connection from the jth to ith neuron; the memory patterns are stored in this connections. U _serepresents the steady state value of the variable u _i(t). The strength of synaptic transmission is given by the product of x _j(t) and u _j(t); the strength decreases (depression) or increases (facilitation) depending on the parameters τ _R, τ _F, and U _se.

The associative memory network is implemented with following absolute strength of synaptic connection $J_{ij} = \frac{1} {N}\sum _{\mu =1}^{p}\xi _{ i}^{\mu }\xi _{ j}^{\mu }$ where J _ii = 0. The p memory patterns $\xi ^{\mu } = (\xi _{1}^{\mu },\cdots \,,\xi _{N}^{\mu }),\mu = 1,\cdots \,,p,\xi _{1}^{\mu } \in \{-1,1\}$ are given by the following correlated patterns. Suppose that a parent memory pattern ξ, which satisfy $\mathrm{Prob}[\xi _{i} = \pm 1] = 1/2$ . The memory patterns ξ ^μare given by $\mathrm{Prob}[\xi _{i}^{\mu } = \pm 1] = (1 \pm b\xi _{i})/2$ , where b is the correlation level among memory patterns and takes values in the interval [0, 1].

To analyze the macroscopic properties of the associative memory network, we consider the following macroscopic mean field model that captures overall dynamical properties of the network [4].

$\displaystyle\begin{array}{rcl} m_{\boldsymbol{\eta }}(t + 1)& =& g_{\beta }\left (\sum _{\boldsymbol{\eta }^{{\prime}}}p_{\boldsymbol{\eta }^{{\prime}}}\boldsymbol{\eta }\cdot \boldsymbol{\eta }^{{\prime}}\left (2m_{\boldsymbol{\eta }^{{\prime} }}(t)X_{\boldsymbol{\eta }^{{\prime}}}(t)U_{\boldsymbol{\eta }^{{\prime}}}(t)/U_{se} - 1\right )\right ),{}\end{array}$