iaf_bw_2001_exact – Leaky integrate-and-fire-neuron model with conductance-based synapses and additional NMDA receptors.

Description

iaf_bw_2001_exact is a leaky integrate-and-fire neuron model with

• an exact implementation of the neuron model described in [1].

• exponential conductance-based AMPA and GABA-synapses

• NMDA synapses with slow nonlinear dynamics

• a fixed refractory period

• no adaptation mechanisms

Neuron and synaptic dynamics

The membrane potential and synaptic variables evolve according to

\[\begin{split}C_\mathrm{m} \frac{dV(t)}{dt} &= -g_\mathrm{L} (V(t) - V_\mathrm{L}) - I_\mathrm{syn} (t) \\[3ex] I_\mathrm{syn}(t) &= I_\mathrm{AMPA}(t) + I_\mathrm{NMDA}(t) + I_\mathrm{GABA}(t) (t) \\[3ex] I_\mathrm{AMPA} &= (V(t) - V_E)\sum_{j \in \Gamma_\mathrm{ex}}^{N_E}w_jS_{j,\mathrm{AMPA}}(t) \\[3ex] I_\mathrm{NMDA} &= \frac{(V(t) - V_E)}{1+[\mathrm{Mg^{2+}}]\mathrm{exp}(-0.062V(t))/3.57}\sum_{j \in \Gamma_\mathrm{ex}}^{N_E}w_jS_{j,\mathrm{NMDA}}(t) \\[3ex] I_\mathrm{GABA} &= (V(t) - V_I)\sum_{j \in \Gamma_\mathrm{in}}^{N_E}w_jS_{j,\mathrm{GABA}}(t) \\[5ex] \frac{dS_{j,\mathrm{AMPA}}}{dt} &=-\frac{j,S_{\mathrm{AMPA}}}{\tau_\mathrm{AMPA}}+\sum_{k \in \Delta_j} \delta (t - t_j^k) \\[3ex] \frac{dS_{j,\mathrm{GABA}}}{dt} &= -\frac{S_{j,\mathrm{GABA}}}{\tau_\mathrm{GABA}} + \sum_{k \in \Delta_j} \delta (t - t_j^k) \\[3ex] \frac{dS_{j,\mathrm{NMDA}}}{dt} &= -\frac{S_{j,\mathrm{NMDA}}}{\tau_\mathrm{NMDA,decay}}+ \alpha x_j (1 - S_{j,\mathrm{NMDA}})\\[3ex] \frac{dx_j}{dt} &= -\frac{x_j}{\tau_\mathrm{NMDA,rise}} + \sum_{k \in \Delta_j} \delta (t - t_j^k)\end{split}\]

where \(\Gamma_\mathrm{ex}\) and \(\Gamma_\mathrm{in}\) are index sets for presynaptic excitatory and inhibitory neurons respectively, and \(\Delta_j\) is an index set for the spike times of neuron \(j\).

Since \(S_{j,\mathrm{AMPA}}\) and \(S_{j,\mathrm{GABA}}\) are piecewise exponential functions, the sums are also a piecewise exponential function, and can be stored in a single synaptic variable each, \(S_{\mathrm{AMPA}}\) and \(S_{\mathrm{GABA}}\) respectively. The sum over \(S_{j,\mathrm{NMDA}}\) does not have a simple expression, and cannot be simplified. Therefore, for each synapse, we need to integrate separate state variables, which makes the model slow.

The specification of this model differs slightly from the one in [1]. The parameters \(g_\mathrm{AMPA}\), \(g_\mathrm{GABA}\), and \(g_\mathrm{NMDA}\) have been absorbed into the respective synaptic weights. Additionally, the synapses from the external population is not separated from the recurrent AMPA-synapses. This model is slow to simulate when there are many neurons with NMDA-synapses, since each post-synaptic neuron simulates each pre-synaptic connection explicitly. The model iaf_bw_2001 is an approximation to this model which is significantly faster.

See also [2], [3]

Parameters

The following parameters can be set in the status dictionary.

Parameter	Default	Math equivalent	Description
E_L	-70.0 mV	\(E_\mathrm{L}\)	Leak reversal potential
E_ex	0.0 mV	\(E_\mathrm{ex}\)	Excitatory reversal potential
E_in	-70.0 mV	\(E_\mathrm{in}\)	Inhibitory reversal potential
V_th	-55.0 mV	\(V_\mathrm{th}\)	Spike threshold
V_reset	-60.0 mV	\(V_\mathrm{reset}\)	Reset potential of the membrane
C_m	250.0 pF	\(C_\mathrm{m}\)	Capacitance of the membrane
g_L	25.0 nS	\(g_\mathrm{L}\)	Leak conductance
t_ref	2.0 ms	\(t_\mathrm{ref}\)	Duration of refractory period
tau_AMPA	2.0 ms	\(\tau_\mathrm{AMPA}\)	Time constant of AMPA synapse
tau_GABA	5.0 ms	\(\tau_\mathrm{GABA}\)	Time constant of GABA synapse
tau_rise_NMDA	2.0 ms	\(\tau_\mathrm{NMDA,rise}\)	Rise time constant of NMDA synapse
tau_decay_NMDA	100.0 ms	\(\tau_\mathrm{NMDA,decay}\)	Decay time constant of NMDA synapse
alpha	0.5 ms^{-1}	\(\alpha\)	Rise-time coupling strength for NMDA synapse
conc_Mg2	1.0 mM	\([\mathrm{Mg}^+]\)	Extracellular magnesium concentration
gsl_error_tol	1e-3		Error tolerance for GSL RKF45-solver

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	\(V_{\mathrm{m}}\)	Membrane potential
s_AMPA	0	\(s_\mathrm{AMPA}\)	AMPA gating variable
s_GABA	0	\(s_\mathrm{GABA}\)	GABA gating variable
s_NMDA	0	\(s_\mathrm{NMDA}\)	NMDA gating variable
I_NMDA	0 pA	\(I_\mathrm{NMDA}\)	NMDA current
I_AMPA	0 pA	\(I_\mathrm{AMPA}\)	AMPA current
I_GABA	0 pA	\(I_\mathrm{GABA}\)	GABA current

Note

It is possible to set values for \(V_\mathrm{m}\), \(S_\mathrm{AMPA}\) and \(S_\mathrm{GABA}\) when creating the model, while the different \(s_{j,\mathrm{NMDA}}\) (j represents presynaptic neuron j) can not be set by the user.

Note

\(g_{\mathrm{\{\{rec,AMPA\}, \{ext,AMPA\}, GABA, NMBA}\}}\) from [1] is built into the weights in this NEST model, so these variables are set by changing the weights.

Sends

SpikeEvent

Receives

SpikeEvent, CurrentEvent, DataLoggingRequest

References

[1] (1,2,3)

Wang, X.-J. (1999). Synaptic Basis of Cortical Persistent Activity: The Importance of NMDA Receptors to Working Memory. Journal of Neuroscience, 19(21), 9587–9603. https://doi.org/10.1523/JNEUROSCI.19-21-09587.1999

[2]

Brunel, N., & Wang, X.-J. (2001). Effects of Neuromodulation in a Cortical Network Model of Object Working Memory Dominated by Recurrent Inhibition. Journal of Computational Neuroscience, 11(1), 63–85. https://doi.org/10.1023/A:1011204814320

[3]

Wang, X. J. (2002). Probabilistic decision making by slow reverberation in cortical circuits. Neuron, 36(5), 955-968. https://doi.org/10.1016/S0896-6273(02)01092-9

See also

iaf_bw_2001

Examples using this model

• Decision making in recurrent network with NMDA-dynamics