iaf_psc_alpha – Leaky integrate-and-fire model with alpha-shaped input currents

Description

iaf_psc_alpha is a leaky integrate-and-fire neuron model with

  • a hard threshold,

  • a fixed refractory period,

  • no adaptation mechanisms,

  • \(\alpha\)-shaped synaptic input currents.

Membrane potential evolution, spike emission, and refractoriness

The membrane potential evolves according to

\[\frac{dV_\text{m}}{dt} = -\frac{V_{\text{m}} - E_\text{L}}{\tau_{\text{m}}} + \frac{I_{\text{syn}} + I_\text{e}}{C_{\text{m}}}\]

where the synaptic input current \(I_{\text{syn}}(t)\) is discussed below and \(I_\text{e}\) is a constant input current set as a model parameter.

A spike is emitted at time step \(t^*=t_{k+1}\) if

\[V_\text{m}(t_k) < V_{th} \quad\text{and}\quad V_\text{m}(t_{k+1})\geq V_\text{th} \;.\]

Subsequently,

\[V_\text{m}(t) = V_{\text{reset}} \quad\text{for}\quad t^* \leq t < t^* + t_{\text{ref}} \;,\]

that is, the membrane potential is clamped to \(V_{\text{reset}}\) during the refractory period.

Synaptic input

The synaptic input current has an excitatory and an inhibitory component

\[I_{\text{syn}}(t) = I_{\text{syn, ex}}(t) + I_{\text{syn, in}}(t)\]

where

\[I_{\text{syn, X}}(t) = \sum_{j} w_j \sum_k i_{\text{syn, X}}(t-t_j^k-d_j) \;,\]

where \(j\) indexes either excitatory (\(\text{X} = \text{ex}\)) or inhibitory (\(\text{X} = \text{in}\)) presynaptic neurons, \(k\) indexes the spike times of neuron \(j\), and \(d_j\) is the delay from neuron \(j\).

The individual post-synaptic currents (PSCs) are given by

\[i_{\text{syn, X}}(t) = \frac{e}{\tau_{\text{syn, X}}} t e^{-\frac{t}{\tau_{\text{syn, X}}}} \Theta(t)\]

where \(\Theta(x)\) is the Heaviside step function. The PSCs are normalized to unit maximum, that is,

\[i_{\text{syn, X}}(t= \tau_{\text{syn, X}}) = 1 \;.\]

As a consequence, the total charge \(q\) transferred by a single PSC depends on the synaptic time constant according to

\[q = \int_0^{\infty} i_{\text{syn, X}}(t) dt = e \tau_{\text{syn, X}} \;.\]

By default, \(V_\text{m}\) is not bounded from below. To limit hyperpolarization to biophysically plausible values, set parameter \(V_{\text{min}}\) as lower bound of \(V_\text{m}\).

Note

NEST uses exact integration [1], [2] to integrate subthreshold membrane dynamics with maximum precision; see also [3].

If \(\tau_\text{m}\approx \tau_{\text{syn, ex}}\) or \(\tau_\text{m}\approx \tau_{\text{syn, in}}\), the model will numerically behave as if \(\tau_\text{m} = \tau_{\text{syn, ex}}\) or \(\tau_\text{m} = \tau_{\text{syn, in}}\), respectively, to avoid numerical instabilities.

For implementation details see the IAF Integration Singularity notebook.

Parameters

The following parameters can be set in the status dictionary.

Parameter

Default

Math equivalent

Description

E_L

-70 mV

\(E_\text{L}\)

Resting membrane potential

C_m

250 pF

\(C_{\text{m}}\)

Capacity of the membrane

tau_m

10 ms

\(\tau_{\text{m}}\)

Membrane time constant

t_ref

2 ms

\(t_{\text{ref}}\)

Duration of refractory period

V_th

-55 mV

\(V_{\text{th}}\)

Spike threshold

V_reset

-70 mV

\(V_{\text{reset}}\)

Reset potential of the membrane

tau_syn_ex

2 ms

\(\tau_{\text{syn, ex}}\)

Rise time of the excitatory synaptic alpha function

tau_syn_in

2 ms

\(\tau_{\text{syn, in}}\)

Rise time of the inhibitory synaptic alpha function

I_e

0 pA

\(I_\text{e}\)

Constant input current

V_min

\(-\infty\) mV

\(V_{\text{min}}\)

Absolute lower value for the membrane potential

The following state variables evolve during simulation and are available either as neuron properties or as recordables.

State variable

Initial value

Math equivalent

Description

V_m

-70 mV

\(V_{\text{m}}\)

Membrane potential

I_syn_ex

0 pA

\(I_{\text{syn, ex}}\)

Excitatory synaptic input current

I_syn_in

0 pA

\(I_{\text{syn, in}}\)

Inhibitory synaptic input current

References

Sends

SpikeEvent

Receives

SpikeEvent, CurrentEvent, DataLoggingRequest

See also

Neuron, Integrate-And-Fire, Current-Based

Examples using this model