The considered neural network consists of a central neuron with state variables (\(u ,v ,\omega\)) and \(m\) neighbor neurons coupled with the central one and having state variables (\(u_i,v_i,\omega _i\)). The dynamics of the network is described by a system of Hindmarsh-Rose equations:
\(\frac{\partial u}{\partial t}=d \Delta u + au^2-bu^3+v-\omega+J,\)
\(\frac{\partial v}{\partial t}=\alpha- v - \beta u^2,\)
\(\frac{\partial \omega}{\partial t}= q (u-c)-r \omega,\)
\(\frac{\partial u_i}{\partial t} = d {\Delta} u_i + a{u_i}^2-b{u_i}^3+v_i-\omega_i+J,\)
\(\frac{\partial v_i}{\partial t} = \alpha-v_i - \beta {u_i}^2,\)
\(\frac{\partial \omega_i}{\partial t}= q (u_i-c)-r \omega_i.\)
for \(i=1,\dots,m\).
The system is subject to initial and coupled boundary conditions with \(t>0\), \(x\in\Omega\subset \mathbb R^n\), \(n\leq 3\). The initial-boundary value problem is formulated as an initial value problem for an evolutionary equation and one proves the existence of a unique global weak solution in time for this problem and its continuous dependence on initial data. If a special threshold condition for stimulation signal strength of the boundary coupled Hindmarsh-Rose neural network is satisfied, a result on the asymptotic synchronization of the neural network is proved.