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6.3 Correlation of Neuronal State Disturbances

The statistical aspects of random intervals between the action potentials of biological neurons are normally decided by the irregularities due to neural conduction velocity/dynamics, axonal fiber type mixture, synchronization/asynchronization effects (arising from dropping out of certain neuron units in the synaptic transmission), percentage of polyphasic action potentials, etc. The temporal dynamics of the neural conduction (or the action potential) can therefore be modeled as a train of delta-Dirac functions (representing a symmetric dichotomous process with bistable values), the time interval between their occurrences being a random variable (Figure 6.2a). Though in a memoryless mathematical neuron model, the relevant statistics of the recurrence of action potentials is presumed to be independent of the other events; the process underlying the neuronal disturbance cannot be altogether assumed as free of dependency on the previous history. As pointed out by McCulloch and Pitts [7], there is a possibility that a particular neural state has dependency at least on the preceding event.


Figure 6.2  Models of action potential train (a) Delta-dirac (impulse) representation; (b) Semirandom telegraphic signal representation (markovian statistics)

In other words, markovian statistics can be attributed to the neuronal state transition and the occurrence of action potentials can be modeled as a symmetric dichotomous Markov process which has bistable values at random intervals. The waiting times in each state are exponentially distributed (which ensures markovian structure of the process involved) having a correlation function given by:

Here, ρ2 = (kBTΓ) and the symmetric dichotomous Markov variable Xt represents the random process whose value switches between two extremes (all-or-none) ±ρ, at random times. The correlation time τc is equal to 1/Γ, and the mean frequency of transition from one value to the other is Γ/2. That is, the stochastic system has two state epochs, namely, random intervals of occurrence and random finite duration of the occurrences. (In a simple delta-Dirac representation, however, the durations of the disturbances are assumed to be of zero values). The process Xt (t = 0) should therefore represent approximately a semirandom telegraphic signal (Figure 6.2b).

The transition probability between the bistable values is dictated by the Chapman-Kolmogorov system of equations [88] for integer-valued variates; and the spectral density of the dichotomous Markov process can be specified by the Fourier transform of the correlation function given by Equation (6.4) and corresponds to the well-known Lorentzian relation given by:

It can be presumed that a synchronism exists between the recursive disturbances, at least on a short-term/quasistationary basis [33]. This could result from more or less simultaneous activation of different sections of presynaptic fibers.


Figure 6.3  Spectral densities of a dichotomous markov process (a): Under aperiodic limit; (b): Under periodic limit

In the delta-Dirac function model, this synchronism is rather absolute and implicit. In the markovian dichotomous model, the synchronism can be inculcated by a periodic attribute or an external parameter so that the periodic variation will mimic the dichotomous Markov process with a correlation time 1/Γ, assuming average switching frequencies of the two variations to be identical. That is, . For this periodic fluctuation, the correlation function is a sawtooth wave between ±ρ2 of fundamental angular frequency 2πν. The corresponding Fourier spectrum is given by:

where δ(ω - 2πqν) is an impulse unit area occurring at frequency ω = 2πqν. Typical normalized spectral densities corresponding to the periodic and aperiodic limits of dichotomous Markov process are depicted in Figure 6.3.

Inasmuch as the output of the neuronal unit has a characteristic colored noise spectrum, it can be surmised that this limited bandwidth of the noise observed at the output should be due to the intrinsic, nonwhite spectral properties of the neuronal disturbances. This is because the firing action at the cell itself would not introduce band-limiting on the intrinsic disturbances. The reason is as follows: The state changing or the time response of the neuronal cell refers to a signum-type switching function or a transient time response as depicted in Figure 6.4.


Figure 6.4  Transient response of a neuronal cell (a): For arbitrary value of α < ∞; (b): For the value of α → ∞

The transient time response f(αt) has a frequency spectrum specified in terms of the Laplace transform given by:

where α is a constant and as α → ∞, the transient response assumes the ideal signum-type switching function in which case the frequency is directly proportional to (1/ω). However, the output of the neuronal unit has (1/ω2) spectral characteristics as could be evinced from Equation (6.5). Therefore, the switching action at the neuronal cell has less influence in dictating the output spectral properties of the noise. In other words, the colored frequency response of the disturbances elucidated at the output should be essentially due to the colored intrinsic/inherent spectral characteristics of the disturbances existing at the neuronal structure.

Hence, in general, it is not justifiable to presume the spectral characteristics of neural disturbances as a flat-band white noise and the solutions of Langevin and/or Fokker-Planck equations described before should therefore correspond to a colored noise situation.


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