iaf_psc_delta – Leaky integrate-and-fire model with delta-shaped input currents

Description

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

• a hard threshold,

• a fixed refractory period,

• Dirac delta (\(\delta\))-shaped synaptic input currents.

Membrane potential evolution, spike emission, and refractoriness

The membrane potential evolves according to

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

where the derivative of change in voltage due to synaptic input \(\dot{\Delta}_{\text{syn}}(t)\) 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 change in membrane potential due to synaptic inputs can be formulated as:

\[\dot{\Delta}_{\text{syn}}(t) = \sum_{j} w_j \sum_k \delta(t-t_j^k-d_j) \;,\]

where \(j\) indexes either excitatory (\(w_j > 0\)) or inhibitory (\(w_j < 0\)) presynaptic neurons, \(k\) indexes the spike times of neuron \(j\), \(d_j\) is the delay from neuron \(j\), and \(\delta\) is the Dirac delta distribution. This implies that the jump in voltage upon a single synaptic input spike is

\[\Delta_{\text{syn}} = w \;,\]

where \(w\) is the corresponding synaptic weight in mV.

The change in voltage caused by the synaptic input can be interpreted as being caused by individual post-synaptic currents (PSCs) given by

\[i_{\text{syn}}(t) = C_{\text{m}} \cdot w \cdot \delta(t).\]

As a consequence, the total charge \(q\) transferred by a single PSC is

\[q = \int_0^{\infty} i_{\text{syn, X}}(t) dt = C_{\text{m}} \cdot w \;.\]

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

Spikes arriving while the neuron is refractory, are discarded by default. If the property refractory_input is set to True, such spikes are added to the membrane potential at the end of the refractory period, dampened according to the interval between arrival and end of refractoriness.

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	\(C_{\text{m}}\)	Capacitance 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
I_e	0 pA	\(I_\text{e}\)	Constant input current
V_min	\(-\infty\) mV	\(V_{\text{min}}\)	Absolute lower value for the membrane potential
refractory_input	False	None	If set to True, spikes arriving during refractory period are integrated afterwards

References

[1]

Rotter S, Diesmann M (1999). Exact simulation of time-invariant linear systems with applications to neuronal modeling. Biologial Cybernetics 81:381-402. DOI: https://doi.org/10.1007/s004220050570

[2]

Diesmann M, Gewaltig M-O, Rotter S, & Aertsen A (2001). State space analysis of synchronous spiking in cortical neural networks. Neurocomputing 38-40:565-571. DOI: https://doi.org/10.1016/S0925-2312(01)00409-X

Sends

SpikeEvent

Receives

SpikeEvent, CurrentEvent, DataLoggingRequest

See also

Neuron, Integrate-And-Fire, Current-Based

Examples using this model

• Comparing precise and grid-based neuron models

• Random balanced network (delta synapses)

• Sensitivity to perturbation

• Store and restore a network simulation

• Synapse Collection usage example