iaf_bw_2001 – Leaky integrate-and-fire-neuron model with conductance-based synapses and additional NMDA receptors with simplified dynamics.

Description

iaf_bw_2001 is a leaky integrate-and-fire neuron model with

• an approximate version of the neuron model described in [1], [2], [3].

• exponential conductance-based AMPA and GABA-synapses

• exponential conductance-based NMDA-synapses weighted such that it approximates the original non-linear dynamics

• a fixed refractory period

• no adaptation mechanisms

Neuron and synaptic dynamics

The membrane potential and synaptic variables evolve according to

\[\begin{split}C_\mathrm{m} \frac{dV(t)}{dt} &= -g_\mathrm{L} (V(t) - V_\mathrm{L}) - I_\mathrm{syn} (t) \\[3ex] I_\mathrm{syn}(t) &= I_\mathrm{AMPA}(t) + I_\mathrm{NMDA}(t) + I_\mathrm{GABA}(t) (t) \\[3ex] I_\mathrm{AMPA} &= (V(t) - V_E)\sum_{j \in \Gamma_\mathrm{ex}}^{N_E}w_jS_{j,\mathrm{AMPA}}(t) \\[3ex] I_\mathrm{NMDA} &= \frac{(V(t) - V_E)}{1+[\mathrm{Mg^{2+}}]\mathrm{exp}(-0.062V(t))/3.57}\sum_{j \in \Gamma_\mathrm{ex}}^{N_E}w_jS_{j,\mathrm{NMDA}}(t) \\[3ex] I_\mathrm{GABA} &= (V(t) - V_I)\sum_{j \in \Gamma_\mathrm{in}}^{N_E}w_jS_{j,\mathrm{GABA}}(t) \\[5ex] \frac{dS_{j,\mathrm{AMPA}}}{dt} &= -\frac{j,S_{\mathrm{AMPA}}}{\tau_\mathrm{AMPA}}+\sum_{k \in \Delta_j} \delta (t - t_j^k) \\[3ex] \frac{dS_{j,\mathrm{GABA}}}{dt} &= -\frac{S_{j,\mathrm{GABA}}}{\tau_\mathrm{GABA}} + \sum_{k \in \Delta_j} \delta (t - t_j^k) \\[3ex] \frac{dS_{j,\mathrm{NMDA}}}{dt} &= -\frac{S_{j,\mathrm{NMDA}}}{\tau_\mathrm{NMDA,decay}} + \sum_{k \in \Delta_j} (k_0 + k_1 S(t)) \delta (t - t_j^k) \\[3ex]\end{split}\]

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 \(j\).

\[\begin{split}k_0 &= (\alpha \tau_r)^{\frac{\tau_r}{\tau_d}} \gamma \big[1 - \frac{\tau_r}{\tau_d}, \alpha \tau_r \big] \\[3ex] k_1 &= \mathrm{exp}(-\alpha \tau_\mathrm{r}) - 1\end{split}\]

where \(\gamma\) is the lower incomplete gamma function. For these values of \(k_0\) and \(k_1\), the approximate model will approach the exact model for large t.

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 are not separated from the recurrent AMPA-synapses.

See also [2] and [3].

For more implementation details and a comparison to the exact version, see:

• Brunel_Wang_2001_Model_Approximation

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

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

Note

For the NMDA dynamics to work, both pre-synaptic and post-synaptic neurons must be of type iaf_bw_2001. For AMPA/GABA synapses, any pre-synaptic neuron can be used.

Note

For technical reasons, spikes from iaf_bw_2001 neurons must be recorded with time_in_steps: True set in the spike recorder, ignoring the offset value. We hope to correct this in a future version of NEST.

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] (1,2)

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] (1,2)

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_exact

Examples using this model

• Decision making in recurrent network with NMDA-dynamics