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 potenial (default \(-\infty\))

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

[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

[3]

Morrison A, Straube S, Plesser H E, Diesmann M (2006). Exact subthreshold integration with continuous spike times in discrete time neural network simulations. Neural Computation, in press DOI: https://doi.org/10.1162/neco.2007.19.1.47

Sends

SpikeEvent

Receives

SpikeEvent, CurrentEvent, DataLoggingRequest

See also

Neuron, Integrate-And-Fire, Current-Based

Examples using this model

• Balanced neuron example

• Campbell & Siegert approximation example

• Comparing precise and grid-based neuron models

• IAF neuron example with current generator

• One neuron example

• One neuron with noise

• Plot weight matrices example

• Pulse packet example

• Random balanced network (alpha synapses) connected with NEST

• Random balanced network HPC benchmark

• Spatial networks: 4x3 grid

• Spatial networks: 4x3 grid for three populations

• Spatial networks: 4x3 grid with pyramidal cells and interneurons

• Spatial networks: A spatial network in 3D

• Spatial networks: A spatial network in 3D with Gaussian connection probabilities

• Spatial networks: A spatial network in 3D with exponential connection probabilities

• Spatial networks: Circular mask and flat probability

• Spatial networks: Circular mask and flat probability, with edge wrap

• Spatial networks: Convergent projection and rectangular mask, from source perspective

• Spatial networks: Convergent projection and rectangular mask, from target perspective

• Spatial networks: Freely placed neurons

• Spatial networks: Gaussian probabilistic kernel

• Spatial networks: Pyramidal cells and interneurons

• Spatial networks: Showcase of PlotTargets, PlotSources, GetTargetNodes, GetSourceNodes

• Spatially-structured networks in NEST

• Spike synchronization through subthreshold oscillation

• Structural Plasticity example

• Synapse Collection usage example

• Two neuron example

• Use evolution strategies to find parameters for a random balanced network (alpha synapses)

• Using CSA for connection setup

• Using CSA with spatial populations