Simulated power spectral density (PSD) of background electrocorticogram (ECoG)

Abstract

The ECoG background activity of cerebral cortex in states of rest and slow wave sleep resembles broadband noise. The power spectral density (PSD) then may often conform to a power-law distribution: a straight line in coordinates of log power vs. log frequency. The exponent, x, of the distribution, 1/f(x), ranges between 2 and 4. These findings are explained with a model of the neural source of the background activity in mutual excitation among pyramidal cells. The dendritic response of a population of interactive excitatory neurons to an impulse input is a rapid exponential rise and a slow exponential decay, which can be fitted with the sum of two exponential terms. When that function is convolved as the kernel with pulses from a Poisson process and summed, the resulting "brown" or "black noise conforms to the ECoG time series and the PSD in rest and sleep. The PSD slope is dependent on the rate of rise. The variation in the observed slope is attributed to variation in the level of the background activity that is homeostatically regulated by the refractory periods of the excitatory neurons. Departures in behavior from rest and sleep to action are accompanied by local peaks in the PSD, which manifest emergent nonrandom structure in the ECoG, and which prevent reliable estimation of the 1/f(x) exponents in active states. We conclude that the resting ECoG truly is low-dimensional noise, and that the resting state is an optimal starting point for defining and measuring both artifactual and physiological structures emergent in the activated ECoG.

Fig. 1

Examples are shown of straight lines fitted to PSD (power/unit frequency in log₁₀ scale) from ECoG in rabbit (a–c) and human (d) in contrasting states of relative behavioral inactivity vs. states of engagement. The lines were fitted by hand and eye in order to calculate the exponent, x, and to emphasize the appearance of peaks above the putative 1/f^x relation. The short line in (d) between 10 and 100 Hz was fitted by linear regression. Adapted from Freeman et al. (2006)

Fig. 2

(a) Two examples of the impulse responses of the bulbar periglomerular mutually excitatory population at lowest and highest stimulus intensities give decay rates respectively of b = 40/s and 90/s and rise rates respectively of 500/s and 300/s. Adapted from Freeman (1974)

Fig. 3

(a) Extrapolation of the relation of response amplitude as a function of stimulus intensity gives the threshold of the impulse response to an excitatory stimulus. (b) Extrapolation of the measured rate constants to threshold gives the rate constant of zero at response amplitude of zero, which is evidence for the set point at which the background activity is stabilized by neurohumoral controls. That regulation is expressed in the limitations on neural firing rates imposed by their refractory periods. Adapted from Freeman (1974)

Fig. 4

Histograms of the slopes of PSD from human ECoG were constructed for the states of awake rest and slow wave sleep. A straight line was fitted b regression to each PSD in the frequency range from 3–100 Hz for calculation of the slope, the intercept, and the residuals expressed as % of the total variance in the designated frequency range. From Freeman et al. (2006)

Fig. 5

(a) Two examples are shown of the impulse response used as the kernel of integration for convolution with the Poisson process: α = 1/s, β = 1000/s or 10/s, step size 2 ms. (b–d) Simulated PSD with the three filter settings illustrating the slopes and the roll-off frequencies

Fig. 6

The exponents x (negative of slope in 1/f^x) depended on rise rate, β, not on decay rate, α. The data were fitted with Eq. 2 derived from the 2-pole filter

Fig. 7

The impulse responses in Fig. 5 are approximated by Eq. 1, with the decay rate and rise rate taken from Table 2 in Freeman (1974). These kernels were convolved with the random pulse trains giving the simulated ECoG, from which the PSD were calculated by the multitaper method for the periodogram