iaf_cond_exp – Leaky integrate-and-fire neuron model with exponentially-shaped synaptic conductances

Description

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

• a hard threshold,

• a fixed refractory period,

• no adaptation mechanisms,

• exponentially-shaped synaptic conductances according to [1], normalized such that an event of weight 1.0 results in a peak conductance of 1 nS.

Membrane potential evolution, spike emission, and refractoriness

The membrane potential evolves according to

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

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

\[I_{\text{syn, X}}(t) = (V_{\text{m}}(t) - E_{\text{syn, X}}) \sum_{j} \sum_k g_{\text{j, X}}(t-t_j^k-d_j) \;,\]

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 synaptic conductances are given by

\[g_{\text{j, X}}(t) = w_{\text{j}} \cdot e^{-\frac{t}{\tau_{\text{syn, X}}}} \Theta(t)\]

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

\[g_{\text{j, X}}(t= 0) = w_{\text{j}} \;,\]

where \(w\) is a weight (excitatory if \(w > 0\) or inhibitory if \(w < 0\)).

Parameters

Parameter	Default	Math equivalent	Description
E_L	-70 mV	\(E_\text{L}\)	Leak reversal potential
C_m	250 pF	\(C_{\text{m}}\)	Capacity of the membrane
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
E_ex	0 mV	\(E_\text{ex}\)	Excitatory reversal potential
E_in	-85 mV	\(E_\text{in}\)	Inhibitory reversal potential
g_L	-16.6667 nS	\(g_\text{L}\)	Leak conductance
tau_syn_ex	0.2 ms	\(\tau_{\text{syn, ex}}\)	Exponential decay time constant of excitatory synaptic conductance kernel
tau_syn_in	2.0 ms	\(\tau_{\text{syn, in}}\)	Exponential decay time constant of inhibitory synaptic conductance kernel
I_e	0 pA	\(I_\text{e}\)	Constant input current

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
g_ex	0 nS	\(g_{\text{ex}}\)	Excitatory synaptic conductance
g_in	0 nS	\(g_{\text{in}}\)	Inhibitory synaptic conductance

Sends

SpikeEvent

Receives

SpikeEvent, CurrentEvent, DataLoggingRequest

References

[1]

Meffin H, Burkitt AN, Grayden DB (2004). An analytical model for the large, fluctuating synaptic conductance state typical of neocortical neurons in vivo. Journal of Computational Neuroscience, 16:159-175. DOI: https://doi.org/10.1023/B:JCNS.0000014108.03012.81

See also

iaf_psc_delta, iaf_psc_alpha, iaf_psc_exp, iaf_cond_alpha, iaf_cond_beta

Examples using this model

• Benchmark 1 of the simulator review (COBA)

• Initial membrane voltage

• Music example