iaf_psc_alpha – Leaky integrate-and-fire model with alpha-shaped input currents =============================================================================== Description +++++++++++ ``iaf_psc_alpha`` is a leaky integrate-and-fire neuron model with * a hard threshold, * a fixed refractory period, * no adaptation mechanisms, * :math:`\alpha`-shaped synaptic input currents. 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_{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. 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) = \frac{e}{\tau_{\text{syn, X}}} t e^{-\frac{t}{\tau_{\text{syn, X}}}} \Theta(t) where :math:`\Theta(x)` is the Heaviside step function. The PSCs are normalized to unit maximum, that is, .. math:: i_{\text{syn, X}}(t= \tau_{\text{syn, X}}) = 1 \;. 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 = e \tau_{\text{syn, X}} \;. By default, :math:`V_\text{m}` is not bounded from below. To limit hyperpolarization to biophysically plausible values, set parameter :math:`V_{\text{min}}` as lower bound of :math:`V_\text{m}`. .. note:: NEST uses exact integration [1]_, [2]_ to integrate subthreshold membrane dynamics with maximum precision; see also [3]_. If :math:`\tau_\text{m}\approx \tau_{\text{syn, ex}}` or :math:`\tau_\text{m}\approx \tau_{\text{syn, in}}`, the model will numerically behave as if :math:`\tau_\text{m} = \tau_{\text{syn, ex}}` or :math:`\tau_\text{m} = \tau_{\text{syn, in}}`, respectively, to avoid numerical instabilities. For implementation details see the `IAF Integration Singularity notebook <../model_details/IAF_Integration_Singularity.ipynb>`_. 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 ``V_min`` :math:`-\infty` mV :math:`V_{\text{min}}` Absolute lower value for the membrane potential =============== ================== =============================== ======================================================================== 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] 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 .. [2] 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 .. [3] Morrison A, Straube S, Plesser H E, Diesmann M (2006). Exact subthreshold integration with continuous spike times in discrete time neural network simulations. Neural Computation, in press DOI: https://doi.org/10.1162/neco.2007.19.1.47 Sends +++++ SpikeEvent Receives ++++++++ SpikeEvent, CurrentEvent, DataLoggingRequest See also ++++++++ :doc:`Neuron `, :doc:`Integrate-And-Fire `, :doc:`Current-Based ` Examples using this model +++++++++++++++++++++++++ .. listexamples:: iaf_psc_alpha