The models in this category describe the relationship between neuronal membrane currents at the input stage and membrane voltage at the output stage. This category includes (generalized) integrate-and-fire models and biophysical models inspired by the work of Hodgkin–Huxley in the early 1950s using an experimental setup that punctured the cell membrane and allowed to force a specific membrane voltage/current.).
Hodgkin–Huxley The
Hodgkin–Huxley model (H&H model) is a model of the relationship between the flow of ionic currents across the neuronal cell membrane and the membrane voltage of the cell. it is also possible to extend it to take into account the evolution of the concentrations (considered fixed in the original model).
Perfect Integrate-and-fire One of the earliest models of a neuron is the perfect integrate-and-fire model (also called non-leaky integrate-and-fire), first investigated in 1907 by
Louis Lapicque. A neuron is represented by its membrane voltage which evolves in time during stimulation with an input current according :I(t)=C \frac{d V(t)}{d t} which is just the time
derivative of the law of
capacitance, . When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold , at which point a
delta function spike occurs and the voltage is reset to its
resting potential, after which the model continues to run. The
firing frequency of the model thus increases linearly without bound as input current increases. The model can be made more accurate by introducing a
refractory period that limits the firing frequency of a neuron by preventing it from firing during that period. For constant input the threshold voltage is reached after an integration time after starting from zero. After a reset, the refractory period introduces a dead time so that the total time until the next firing is . The firing frequency is the inverse of the total inter-spike interval (including dead time). The firing frequency as a function of a constant input current, is therefore :\,\! f(I)= \frac{I} {C_\mathrm{} V_\mathrm{th} + t_\mathrm{ref} I}. A shortcoming of this model is that it describes neither adaptation nor leakage. If the model receives a below-threshold short current pulse at some time, it will retain that voltage boost forever - until another input later makes it fire. This characteristic is not in line with observed neuronal behavior. The following extensions make the integrate-and-fire model more plausible from a biological point of view.
Leaky integrate-and-fire The leaky integrate-and-fire model, which can be traced back to
Louis Lapicque, The model can also be used for inhibitory neurons. The most significant disadvantage of this model is that it does not contain neuronal adaptation, so that it cannot describe an experimentally measured spike train in response to constant input current. This disadvantage is removed in generalized integrate-and-fire models that also contain one or several adaptation-variables and are able to predict spike times of cortical neurons under current injection to a high degree of accuracy.
Adaptive integrate-and-fire Neuronal adaptation refers to the fact that even in the presence of a constant current injection into the soma, the intervals between output spikes increase. An adaptive integrate-and-fire neuron model combines the leaky integration of voltage with one or several adaptation variables (see Chapter 6.1. in the textbook Neuronal Dynamics) : \tau_\mathrm{m} \frac{d V_\mathrm{m} (t)}{d t} = R I(t)- [V_\mathrm{m} (t) - E_\mathrm{m} ]- R \sum_k w_k : \tau_k \frac{d w_k (t)}{d t} = - a_k [V_\mathrm{m} (t) - E_\mathrm{m} ]- w_k + b_k \tau_k \sum_f \delta (t-t^f) where \tau_m is the membrane
time constant, is the adaptation current number, with index
k, \tau_k is the time constant of adaptation current , is the resting potential and is the firing time of the neuron and the Greek delta denotes the Dirac delta function. Whenever the voltage reaches the firing threshold the voltage is reset to a value below the firing threshold. The reset value is one of the important parameters of the model. The simplest model of adaptation has only a single adaptation variable and the sum over k is removed. File:Spike Time Prediction with Generalized Integrate-and-Fire model.jpg|thumb|Spike times and subthreshold voltage of cortical neuron models can be predicted by generalized integrate-and-fire models such as the adaptive integrate-and-fire model, the adaptive exponential integrate-and-fire model, or the spike response model. In the example here, adaptation is implemented by a dynamic threshold which increases after each spike. Moreover, adaptive integrate-and-fire neurons with several adaptation variables are able to predict spike times of cortical neurons under time-dependent current injection into the soma. An advantage of this model is that it can capture adaptation effects with a single variable. The model has the following form spike generation is exponential, following the equation: : \frac{dV}{dt} - \frac{R} {\tau_m} I(t)= \frac{1} {\tau_m} \left[ E_m-V+\Delta_T \exp \left( \frac{V - V_T} {\Delta_T} \right) \right]. where V is the membrane potential, V_T is the intrinsic membrane potential threshold, \tau_m is the membrane time constant, E_m is the resting potential, and \Delta_T is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons. In numerical simulation the integration is stopped if the membrane potential hits an arbitrary threshold (much larger than V_T) at which the membrane potential is reset to a value . The voltage reset value is one of the important parameters of the model. Importantly, the right-hand side of the above equation contains a nonlinearity that can be directly extracted from experimental data. the above exponential nonlinearity of the voltage equation is combined with an adaptation variable w : \tau_m \frac{dV}{dt} = R I(t) + \left[ E_m-V+\Delta_T \exp \left( \frac{V - V_T} {\Delta_T} \right) \right] - R w : \tau \frac{d w (t)}{d t} = - a [V_\mathrm{m} (t) - E_\mathrm{m} ]- w + b \tau \delta (t-t^f) , v_{th}, gradually returns to its steady state value depending on the threshold adaptation time constant \tau_{\theta}. This is one of the simpler techniques to achieve spike frequency adaptation. The expression for the adaptive threshold is given by: v_{th}(t) = v_{th0} + \frac{\sum \theta(t - t_f)}{f} = v_{th0} + \frac{\sum \theta_0 \exp\left[-\frac{(t - t_f)}{\tau_{\theta}}\right]}{f} where \theta(t) is defined by: \theta(t) = \theta_0 \exp\left[-\frac{t}{\tau_{\theta}}\right] When the membrane potential, u(t), reaches a threshold, it is reset to v_{rest}: u(t) \geq v_{th}(t) \Rightarrow v(t) = v_{\text{rest}} A simpler version of this with a single time constant in threshold decay with an LIF neuron is realized in to achieve LSTM like recurrent spiking neural networks to achieve accuracy nearer to ANNs on few spatio temporal tasks.
Double Exponential Adaptive Threshold (DEXAT) The DEXAT neuron model is a flavor of adaptive neuron model in which the threshold voltage decays with a double exponential having two time constants. Double
exponential decay is governed by a fast initial decay and then a slower decay over a longer period of time. This neuron used in SNNs through surrogate gradient creates an adaptive learning rate yielding higher accuracy and faster convergence, and flexible
long short-term memory compared to existing counterparts in the literature. The membrane potential dynamics are described through equations and the threshold adaptation rule is: v_{th}(t) = b_{0} + \beta_{1}b_{1}(t) + \beta_{2}b_{2}(t) The dynamics of b_{1}(t) and b_2(t) are given by b_{1}(t + \delta t) = p_{j1}b_{1}(t) + (1 - p_{j1})z(t)\delta(t), b_{2}(t + \delta t) = p_{j2}b_{2}(t) + (1 - p_{j2})z(t)\delta(t), where p_{j1} = \exp\left[-\frac{\delta t}{\tau_{b1}}\right] and p_{j2} = \exp\left[-\frac{\delta t}{\tau_{b2}}\right]. Further, multi-time scale adaptive threshold neuron model showing more complex dynamics is shown in. == Stochastic models of membrane voltage and spike timing ==