Figure 1.
Schematic illustrations of the CMFE and neuronal responses.
(A) We regard a neuron as a stochastic spike generator to produce a random spike sequence, with ISIs determined by a conditional probability distribution for given rate parameter. The value of ξ is determined by synaptic input. (B) The CMFE solves a trade-off between the average energy consumption and the uncertainty of output spike trains at given firing rate ξ, where the former may be proportional to ξ and the latter to −logξ (upper). See Equations 9 and 10. The rate distribution is determined from the balance between the two (lower).
Figure 2.
Power-law inter-spike interval histograms of in vivo cortical neurons.
Juxta-cellular visualization and double-logarithmic plots of the ISI histograms (blue curves) of pyramidal neurons (A) and fast-spiking interneurons (B) recorded in cortical layers 3, 4, 5 and 6. The plots were fitted by neuron-dependent beta-2 distributions (red curves). The four neurons in (B) expressed parvalbumin (PV), a fast-spiking interneuron specific marker (blue: PV, green: biocytin or Neurobiotin). Inset of each panel represents the ISI distributions constructed from the 1st (black) and 2nd (green) halves of the same spike train. (C) Linear regression of the tail of the ISI histogram for one of the 8 neurons shown in (A) (η = 2.91, c.d. = 0.99). (D) The power-law exponents were calculated by linear regression for pyramidal (triangles) and fast-spiking (circles) neurons recorded at various depths of the sensorimotor cortex. Colors indicate the movement components represented by the individual neurons: hold (magenta), pre-movement (green), movement (blue), movement-off (red), post-movement (yellow) and non-related (cyan).
Figure 3.
Estimation of the firing rate in cortical neurons.
(A) The value of κ ( = 3.49) was estimated by the method used in Miura et al. (2007) [35] and the instantaneous firing rate was estimated for a cortical neuron according to the method proposed in Koyama and Shinomoto (2005) [34]. (B) Double-logarithmic histogram of ISIs constructed for the spike train shown in (A) exhibited a power-law decaying tail (black). The histogram was fitted with a beta-2 distribution (gray). (C) Distribution of the estimated instantaneous firing rate (black) was fitted with a gamma distribution (gray). (D) The semi-logarithmic plot of the same rate distribution exhibits an exponentially decaying tail (black) characteristic to the gamma distribution (gray).
Figure 4.
Neuron-dependence of parameter values of the double gamma process.
(A)–(C) Optimal values of beta-2 fitting parameters α, R and κ for pyramidal (triangles) and fast-spiking (circles) neurons recorded at various depths of the cortex. As in Figure 2, the color code indicates the motor components represented by the individual neurons. The values were determined by nonlinear least square method (Materials and Methods). (D) The relationship between α and κ values over the recorded neural population. (E) The relationship between α and η values. The latter was determined by linear regression of the power-law tails of ISI distributions (see Figure 2C).
Figure 5.
Information measures for cortical neurons.
(A, B) The conditional response entropy and entropy of ISI distribuions, respectively, for pyramidal (triangles) and fast-spiking (circles) neurons recorded at various depths of the cortex. Colors indicate the motor component represented by each neuron (see the legend of Figure 2D). (C) Correlations between the conditional response entropy and mutual information for the individual neurons.