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 :math:`\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 .. math:: \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 :math:`I_{\text{syn}}(t)` is discussed below and :math:`I_\text{e}` is a constant input current set as a model parameter. A spike is emitted at time step :math:`t^*=t_{k+1}` if .. math:: V_\text{m}(t_k) < V_{\text{th}} \quad\text{and}\quad V_\text{m}(t_{k+1})\geq V_\text{th} \;. Subsequently, .. math:: 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 :math:`V_{\text{reset}}` during the refractory period. .. note:: Spiking in this model can be either deterministic (:math:`\delta=0`) or stochastic (:math:`\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 .. math:: \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 .. math:: I_{\text{syn}}(t) = I_{\text{syn, ex}}(t) + I_{\text{syn, in}}(t) where .. math:: I_{\text{syn, X}}(t) = \sum_{j} w_j \sum_k i_{\text{syn, X}}(t-t_j^k-d_j) \;, where :math:`j` indexes either excitatory (:math:`\text{X} = \text{ex}`) or inhibitory (:math:`\text{X} = \text{in}`) presynaptic neurons, :math:`k` indexes the spike times of neuron :math:`j`, and :math:`d_j` is the delay from neuron :math:`j`. The individual post-synaptic currents (PSCs) are given by .. math:: i_{\text{syn, X}}(t) = w \cdot e^{-\frac{t}{\tau_{\text{syn, X}}}} \cdot \Theta(t) where :math:`w` is a weight (excitatory if :math:`w > 0` or inhibitory if :math:`w < 0`), and :math:`\Theta(x)` is the Heaviside step function. The time dependent components of the PSCs are normalized to unit maximum, so that, .. math:: i_{\text{syn, X}}(t= 0) = w \;. As a consequence, the total charge :math:`q` transferred by a single PSC depends on the synaptic time constant according to .. math:: 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 <../model_details/IAF_Integration_Singularity.ipynb>`_. ``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 :doc:`PyNEST Microcircuit: Helper Functions <../auto_examples/Potjans_2014/helpers>`. Parameters ++++++++++ The following parameters can be set in the status dictionary. =============== ================== =============================== ======================================================================== **Parameter** **Default** **Math equivalent** **Description** =============== ================== =============================== ======================================================================== ``E_L`` -70 mV :math:`E_\text{L}` Resting membrane potential ``C_m`` 250 pF :math:`C_{\text{m}}` Capacity of the membrane ``tau_m`` 10 ms :math:`\tau_{\text{m}}` Membrane time constant ``t_ref`` 2 ms :math:`t_{\text{ref}}` Duration of refractory period ``V_th`` -55 mV :math:`V_{\text{th}}` Spike threshold ``V_reset`` -70 mV :math:`V_{\text{reset}}` Reset potential of the membrane ``tau_syn_ex`` 2 ms :math:`\tau_{\text{syn, ex}}` Rise time of the excitatory synaptic alpha function ``tau_syn_in`` 2 ms :math:`\tau_{\text{syn, in}}` Rise time of the inhibitory synaptic alpha function ``I_e`` 0 pA :math:`I_\text{e}` Constant input current ``delta`` 0 mV :math:`\delta` Parameter scaling stochastic spiking ``rho`` 0.01 1/s :math:`\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 :math:`V_{\text{m}}` Membrane potential ``I_syn_ex`` 0 pA :math:`I_{\text{syn, ex}}` Excitatory synaptic input current ``I_syn_in`` 0 pA :math:`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 +++++++++++++++++++++++++ .. listexamples:: iaf_psc_exp