I need help in solving a set of equations.
Background: I'd like to model the firing rate of an integrate-and-fire neuron depending on the supply of usable energy that is available. (I don't know how biologically plausible this model is, but first I'd like to solve it.)
The variables (all of them time-dependent):
$\nu$: the firing rate of the neuron
$E$: the supply of usable energy (e.g. to drive the ion pumps)
$P$: the production rate of usable energy
$L$: the loss rate of usable energy
Some constants (depending on the physiology of the neuron):
$\nu_{max}$: the maximal firing rate (e.g. 100Hz)
$E_{min}$: the minimal energy supply that is needed to maintain $\nu_{max}$
$P_0$: base production rate
The initial conditions:
$E_0 > E_{min}$, i.e. the energy store is full
$\nu_0 = \nu_{max}$, i.e. the neuron fires maximally
Important note: The neuron is - from the beginning - exposed to a constant stimulus (e.g. an current injected near the axon hillock) which is strong enough to evoke maximal firing.
Now to the set of equations I'd like to solve:
$\nu(E) = \nu_{max} \min\{1,E/E_{min}\}$ (without energy no firing is possible)
$\dot{E} = P - L$
$L = \lambda\ \nu$
$P = P_0 + p\ \nu$
I am looking for the firing rate $\nu$ and the energy supply $E$ as explicit functions of time, depending on the parameters $E_{min}$, $P_0$, $p$, and $\lambda$, where $p < \lambda$ is assumed.
Is the problem well-posed?
Any hint to which method (better: tool) to use is welcome!
(The first part of the curves is easy: the $\nu$ curve is constantly at $\nu_{max}$, the $E$ curve decreases linearly with constant negative slope $\dot{E} = P_0 + (p - \lambda)\nu_{max}$ (i.e. $P_0 < (\lambda - p)\nu_{max}$ is required). When $E$ reaches $E_{min}$, something else starts to happen, possibly a synchronized oscillation of $\nu$ and $E$ around each other. But what about the amplitudes and frequency of these oscillations in the long-term limit?)
Thanks to Evgeny's comment, the question can be put much simpler: What are the solutions of
$$\dot{E} = \alpha - \beta\min\{1,\gamma E\}$$
with $\alpha < \beta$?