iaf_psc_exp – Leaky integrate-and-fire neuron model with exponential-shaped input currents

Description

iaf_psc_exp a leaky integrate-and-fire model with

• a hard threshold (if \(\delta=0\), see below)

• a fixed refractory period,

• no adaptation mechanisms,

• exponential-shaped synaptic input currents according to [1].

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

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

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.

Note

Spiking in this model can be either deterministic (\(\delta=0\)) or stochastic (\(\delta > 0\)). In the stochastic case, this model implements a type of spike response model with escape noise. Spiking is given by an inhomogeneous Poisson process with rate

\[\rho \exp \left( \frac{V_{\text{m}} - V_{\text{th}}}{\delta} \right).\]

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

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

where \(w\) is a weight (excitatory if \(w > 0\) or inhibitory if \(w < 0\)), and \(\Theta(x)\) is the Heaviside step function. The time dependent components of the PSCs are normalized to unit maximum, so that,

\[i_{\text{syn, X}}(t= 0) = w \;.\]

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

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

Note

If tau_m is very close to tau_syn_ex or tau_syn_in, the model will numerically behave as if tau_m is equal to tau_syn_ex or tau_syn_in, respectively, to avoid numerical instabilities.

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

For implementation details see the IAF Integration Singularity notebook.

iaf_psc_exp can handle current input in two ways:

1. Current input through receptor_type 0 is handled as a stepwise constant current input as in other iaf models, that is, this current directly enters the membrane potential equation.

2. In contrast, current input through receptor_type 1 is filtered through an exponential kernel with the time constant of the excitatory synapse, tau_syn_ex.

   For an example application, see [4].

   Warning: this current input is added to the state variable i_syn_ex_. If this variable is being recorded, its numerical value will thus not correspond to the excitatory synaptic input current, but to the sum of excitatory synaptic input current and the contribution from receptor type 1 currents.

For conversion between postsynaptic potentials (PSPs) and PSCs, please refer to the postsynaptic_potential_to_current function in PyNEST Microcircuit: Helper Functions.

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
delta	0 mV	\(\delta\)	Parameter scaling stochastic spiking
rho	0.01 1/s	\(\rho\)	Baseline stochastic spiking

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]

Tsodyks M, Uziel A, Markram H (2000). Synchrony generation in recurrent networks with frequency-dependent synapses. The Journal of Neuroscience, 20,RC50:1-5. URL: https://infoscience.epfl.ch/record/183402

[2]

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

[3]

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

[4]

Schuecker J, Diesmann M, Helias M (2015). Modulated escape from a metastable state driven by colored noise. Physical Review E 92:052119 DOI: https://doi.org/10.1103/PhysRevE.92.052119

Sends

SpikeEvent

Receives

SpikeEvent, CurrentEvent, DataLoggingRequest

See also

iaf_cond_exp, iaf_psc_exp_ps

Examples using this model

• Class encapsulating the network for solving Sudoku

• Classes to encapsulate the neuronal networks.

• Comparing precise and grid-based neuron models

• Example for the quantal_stp_synapse

• Example of the tsodyks2_synapse in NEST

• PyNEST EI-clustered network: Network Class

• PyNEST EI-clustered network: Network Parameters

• PyNEST Microcircuit: Network Parameters

• Random balanced network (exp synapses, multiple time constants)

• Recording examples

• Short-term depression example

• Short-term facilitation example

• Structural Plasticity example