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and-re (or theta-) neurons (g(V ) = V
2
+ I) [39, 43, 71, 114], abstract neural
oscillator models where U() = b
1
ln(1 + (exp(b)1)) [36, 108], and exponential
IF neurons where g(V ) = IV +exp(V ) [42, 43, 129]. The model class is limited
by the idealization that the sub-threshold neuron dynamics is well characterized by
a single variable; moreover, whereas an analytic approach is conceptually simple
in the limit of innitely fast response (where the kernel is a delta-distribution),
it typically becomes restricted for certain post-synaptic response kernels that are
temporally extended.
13.3.2. Related models
Besides the standard model class dened via (13.1), several variants are widely used
as well. Often, additional degrees of freedom are introduced. One phenomenological
class of models has a spike-triggered adaptation variable [22, 70, 71, 77]. In depen-
dence of the parameters, neurons of this class show a wide spectrum of qualitative
features observed in biological neurons and at the same time allow fast numerical
simulations of large networks if the individual neurons' features are well under-
stood [70, 71]. It has the disadvantages that its dynamics is analytically accessible
only in rare special cases, and in numerical simulations, though they are much faster
than, say for Hodgkin Huxley neurons (see below), the dynamical parameters are
similarly hard to restrict. The `spike response model' works in the original potential
representation and includes additional refractoriness or adaptation, modeled as a
threshold dynamics that is not present in (13.1). Recent works [22, 76, 77] suggest
that certain representatives of spike response model neurons with adaptation well
reproduce the response of real neurons to specic random current inputs.
Spiking neural network models with temporally extended interactions often also
characterize the response dynamics by one additional degree of freedom per neu-
ron, e.g. by a second dierential equation, which is, however, usually chosen to be
solvable in closed form such that Eq. (13.1) is regained (see, e.g., [3, 154, 168]).
Biophysically more detailed models, such as the Hodgkin-Huxley, Morris-Lecar,
Fitzhugh-Nagumo, or Hindmarsh-Rose models ([64, 65, 111], see [71] for a compre-
hensive review) require several dynamical variables and many physiological parame-
ters for each neuron. As such they are appropriate for modeling dynamical network
aspects of well-known systems (see, e.g., [35]); at the same time, they typically
preclude analytical arguments and for many systems it is unclear how to suitably
restrict all model parameters or even whether the chosen model is appropriate at
all [115].
Whereas all deterministic models have their variants that include additional
stochastic inuences, modeling, e.g., synaptic failure [80] or local noise induced by
ion channels [15], intrinsically stochastic models may sometimes be more appropri-
ate for the description of single neuron or network dynamics [49, 78, 88, 127, 128,
143, 144, 156]. For instance, Levina et al. [88, 89] have recently shown that the
dynamics of branching processes under certain conditions well describe the stochas-
