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 .. math:: 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) where :math:`\Gamma_\mathrm{ex}` and :math:`\Gamma_\mathrm{in}` are index sets for presynaptic excitatory and inhibitory neurons respectively, and :math:`\Delta_j` is an index set for the spike times of neuron :math:`j`. Since :math:`S_{j,\mathrm{AMPA}}` and :math:`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, :math:`S_{\mathrm{AMPA}}` and :math:`S_{\mathrm{GABA}}` respectively. The sum over :math:`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 :math:`g_\mathrm{AMPA}`, :math:`g_\mathrm{GABA}`, and :math:`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 :doc:`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 :math:`E_\mathrm{L}` Leak reversal potential ``E_ex`` 0.0 mV :math:`E_\mathrm{ex}` Excitatory reversal potential ``E_in`` -70.0 mV :math:`E_\mathrm{in}` Inhibitory reversal potential ``V_th`` -55.0 mV :math:`V_\mathrm{th}` Spike threshold ``V_reset`` -60.0 mV :math:`V_\mathrm{reset}` Reset potential of the membrane ``C_m`` 250.0 pF :math:`C_\mathrm{m}` Capacitance of the membrane ``g_L`` 25.0 nS :math:`g_\mathrm{L}` Leak conductance ``t_ref`` 2.0 ms :math:`t_\mathrm{ref}` Duration of refractory period ``tau_AMPA`` 2.0 ms :math:`\tau_\mathrm{AMPA}` Time constant of AMPA synapse ``tau_GABA`` 5.0 ms :math:`\tau_\mathrm{GABA}` Time constant of GABA synapse ``tau_rise_NMDA`` 2.0 ms :math:`\tau_\mathrm{NMDA,rise}` Rise time constant of NMDA synapse ``tau_decay_NMDA`` 100.0 ms :math:`\tau_\mathrm{NMDA,decay}` Decay time constant of NMDA synapse ``alpha`` 0.5 ms^{-1} :math:`\alpha` Rise-time coupling strength for NMDA synapse ``conc_Mg2`` 1.0 mM :math:`[\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 :math:`V_{\mathrm{m}}` Membrane potential ``s_AMPA`` 0 :math:`s_\mathrm{AMPA}` AMPA gating variable ``s_GABA`` 0 :math:`s_\mathrm{GABA}` GABA gating variable ``s_NMDA`` 0 :math:`s_\mathrm{NMDA}` NMDA gating variable ``I_NMDA`` 0 pA :math:`I_\mathrm{NMDA}` NMDA current ``I_AMPA`` 0 pA :math:`I_\mathrm{AMPA}` AMPA current ``I_GABA`` 0 pA :math:`I_\mathrm{GABA}` GABA current ================== ================= ========================== ================================= .. note:: It is possible to set values for :math:`V_\mathrm{m}`, :math:`S_\mathrm{AMPA}` and :math:`S_\mathrm{GABA}` when creating the model, while the different :math:`s_{j,\mathrm{NMDA}}` (`j` represents presynaptic neuron `j`) can not be set by the user. .. note:: :math:`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] 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 +++++++++++++++++++++++++ .. listexamples:: iaf_bw_2001_exact