Izhikevich¶

class Izhikevich(shape: tuple[int, ...] | int, step_time: float, *, rest_v: float, crit_v: float, affinity: float, reset_v: float, thresh_v: float, refrac_t: float, tc_membrane: float, tc_adaptation: float | tuple[float, ...], voltage_coupling: float | tuple[float, ...], spike_increment: float | tuple[float, ...], resistance: float = 1.0, batch_size: int = 1, batch_reduction: Callable[[Tensor, tuple[int, ...]], Tensor] | None = None)[source]¶

Bases: AdaptiveCurrentMixin, VoltageMixin, SpikeRefractoryMixin, InfernoNeuron

Simulation of Izhikevich (adaptive quadratic) neuron dynamics.

Implemented as an approximation using Euler’s method.

\[\begin{split}\begin{align*} V_m(t + \Delta t) &= \frac{\Delta t}{\tau_m} \left[ a \left(V_m(t) - V_\text{rest}\right)\left(V_m(t) - V_\text{crit}\right) + R_mI(t) \right] + V_m(t) \\ I(t) &= I_x(t) - \sum_k w_k(t) \\ w_k(t + \Delta t) &= \frac{\Delta t}{\tau_k}\left[ b_k \left[ V_m(t) - V_\text{rest} \right] - w_k(t) \right] + w_k(t) \end{align*}\end{split}\]

If a spike was generated at time \(t\), then.

\[\begin{split}\begin{align*} V_m(t) &\leftarrow V_\text{reset} \\ w_k(t) &\leftarrow w_k(t) + d_k \end{align*}\end{split}\]

Parameters:

shape (tuple[int, ...] | int) – shape of the group of neurons being simulated.
step_time (float) – length of a simulation time step, \(\Delta t\), in \(\text{ms}\).
rest_v (float) – membrane potential difference at equilibrium, \(V_\text{rest}\), in \(\text{mV}\).
crit_v (float) – membrane potential difference at which potential naturally increases, \(V_\text{crit}\), in \(\text{mV}\).
affinity (float) – controls the strength of the membrane potential’s drift towards \(V_\text{rest}\) and away from \(V_\text{crit}\), \(a\), unitless.
reset_v (float) – membrane voltage after an action potential is generated, \(V_\text{reset}\), in \(\text{mV}\).
thresh_v (float) – membrane voltage at which action potentials are generated, in \(\text{mV}\).
refrac_t (float) – length the absolute refractory period, in \(\text{ms}\).
tc_membrane (float) – time constant of exponential decay for membrane voltage, \(\tau_m\), in \(\text{ms}\).
tc_adaptation (float | tuple[float, ...]) – time constant of exponential decay for threshold adaptations, \(\tau_k\), in \(\text{ms}\).
voltage_coupling (float | tuple[float, ...]) – strength of coupling to membrane voltage, \(b_k\), in \(\mu\text{S}\).
spike_increment (float | tuple[float, ...]) – amount by which the adaptive current is increased after a spike, \(d_k\), in \(\text{nA}\).
resistance (float, optional) – resistance across the cell membrane, \(R_m\), in \(\text{M}\Omega\). Defaults to 1.0.
batch_size (int, optional) – size of input batches for simulation. Defaults to 1.

Note

batch_reduction can be one of the functions in PyTorch including but not limited to torch.sum(), torch.mean(), and torch.amax(). A custom function with similar behavior can also be passed in. Like with the included function, it should not keep the original dimensions by default.