  Most of the calcium in the body is stored in bone. The rest is stored
elsewhere, and calcium signalling is one of the most important mechanisms of
information propagation in the body. Yet, many questions remain open. In this
work, we initially consider the mathematical model proposed in Atri et al.
\cite{ atri1993single}. Omitting diffusion, the model is a system of two
nonlinear ordinary differential equations (ODEs) for the calcium concentration,
and the fraction of $IP_3$ receptors that have not been inactivated by the
calcium. We analyse in detail the system as the $IP_3$ concentration, the
\textit{bifurcation parameter}, increases presenting some new insights. We
analyse asymptotically the relaxation oscillations of the model by exploiting a
separation of timescales. Furthermore, motivated by experimental evidence that
cells release calcium when mechanically stimulated and that, in turn, calcium
release affects the mechanical behaviour of cells, we propose an extension of
the Atri model to a 3D nonlinear ODE mechanochemical model, where the
additional equation, derived consistently from a full viscoelastic
\emph{ansatz}, models the evolution of cell/tissue dilatation. Furthermore, in
the calcium dynamics equation we introduce a new "stretch-activation" source
term that induces calcium release and which involves a new bifurcation
parameter, the "strength" of the source. Varying the two bifurcation
parameters, we analyse in detail the interplay of the mechanical and the
chemical effects, and we find that as the strength of the mechanical stimulus
is increased, the $IP_3$ parameter range for which oscillations emerge
decreases, until oscillations eventually vanish at a critical value. Finally,
we analyse the model when the calcium dynamics are assumed faster than the
dynamics of the other two variables.
