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 .. 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}} + \sum_{k \in \Delta_j} (k_0 + k_1 S(t)) \delta (t - t_j^k) \\[3ex] 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`. .. math:: 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 where :math:`\gamma` is the `lower incomplete gamma function `_. For these values of :math:`k_0` and :math:`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 :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 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 <../model_details/Brunel_Wang_2001_Model_Approximation.ipynb>`_ 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:: :math:`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] 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_exact Examples using this model +++++++++++++++++++++++++ .. listexamples:: iaf_bw_2001