Original Paper

Biological Cybernetics

, Volume 104, Issue 6, pp 369-383

First online:

Neuronal model with distributed delay: analysis and simulation study for gamma distribution memory kernel

  • KarmeshuAffiliated withJawaharlal Nehru University Email author 
  • , Varun GuptaAffiliated withUniversity of Texas at Dallas
  • , K. V. KadambariAffiliated withJawaharlal Nehru University

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A single neuronal model incorporating distributed delay (memory)is proposed. The stochastic model has been formulated as a Stochastic Integro-Differential Equation (SIDE) which results in the underlying process being non-Markovian. A detailed analysis of the model when the distributed delay kernel has exponential form (weak delay) has been carried out. The selection of exponential kernel has enabled the transformation of the non-Markovian model to a Markovian model in an extended state space. For the study of First Passage Time (FPT) with exponential delay kernel, the model has been transformed to a system of coupled Stochastic Differential Equations (SDEs) in two-dimensional state space. Simulation studies of the SDEs provide insight into the effect of weak delay kernel on the Inter-Spike Interval(ISI) distribution. A measure based on Jensen–Shannon divergence is proposed which can be used to make a choice between two competing models viz. distributed delay model vis-á-vis LIF model. An interesting feature of the model is that the behavior of (CV(t))(ISI) (Coefficient of Variation) of the ISI distribution with respect to memory kernel time constant parameter η reveals that neuron can switch from a bursting state to non-bursting state as the noise intensity parameter changes. The membrane potential exhibits decaying auto-correlation structure with or without damped oscillatory behavior depending on the choice of parameters. This behavior is in agreement with empirically observed pattern of spike count in a fixed time window. The power spectral density derived from the auto-correlation function is found to exhibit single and double peaks. The model is also examined for the case of strong delay with memory kernel having the form of Gamma distribution. In contrast to fast decay of damped oscillations of the ISI distribution for the model with weak delay kernel, the decay of damped oscillations is found to be slower for the model with strong delay kernel.


Gamma distribution memory kernel Weak and strong delay Exponential distributed delay First passage time ISI distribution Coefficient of variation Autocorrelation function Power spectral density Jensen–Shannon divergence