Network Amplification of Local Fluctuations Causes High Spike Rate Variability, Fractal Firing Patterns and Oscillatory Local Field Potentials

We investigate a model for neural activity in a two-dimensional sheet of leaky integrate-and-fire neurons with feedback connectivity consisting of local excitation and surround inhibition. Each neuron receives stochastic input from an external source, independent in space and time. As recently suggested by Softky and Koch (1992, 1993), independent stochastic input alone cannot explain the high interspike interval variability exhibited by cortical neurons in behaving monkeys. We show that high variability can be obtained due to the amplification of correlated fluctuations in a recurrent network. Furthermore, the cross-correlation functions have a dual structure, with a sharp peak on top of a much broader hill. This is due to the inhibitory and excitatory feedback connections, which cause "hotspots" of neural activity to form within the network. These localized patterns of excitation appear as clusters or stripes that coalesce, disintegrate, or fluctuate in size while simultaneously moving in a random walk constrained by the interaction with other clusters. The synaptic current impinging upon a single neuron shows large fluctuations at many time scales, leading to a large coefficient of variation (C_V) for the interspike interval statistics. The power spectrum associated with single units shows a 1/f decay for small frequencies and is flat at higher frequencies, while the power spectrum of the spiking activity averaged over many cells—equivalent to the local field potential—shows no 1/f decay but a prominent peak around 40 Hz, in agreement with data recorded from cat and monkey cortex (Gray et al. 1990; Eckhorn et al. 1993). Firing rates exhibit self-similarity between 20 and 800 msec, resulting in 1/f-like noise, consistent with the fractal nature of neural spike trains (Teich 1992)

Power-Law Inter-Spike Interval Distributions Infer a Conditional Maximization of Entropy in Cortical Neurons

The brain is considered to use a relatively small amount of energy for its efficient information processing. Under a severe restriction on the energy consumption, the maximization of mutual information (MMI), which is adequate for designing artificial processing machines, may not suit for the brain. The MMI attempts to send information as accurate as possible and this usually requires a sufficient energy supply for establishing clearly discretized communication bands. Here, we derive an alternative hypothesis for neural code from the neuronal activities recorded juxtacellularly in the sensorimotor cortex of behaving rats. Our hypothesis states that in vivo cortical neurons maximize the entropy of neuronal firing under two constraints, one limiting the energy consumption (as assumed previously) and one restricting the uncertainty in output spike sequences at given firing rate. Thus, the conditional maximization of firing-rate entropy (CMFE) solves a tradeoff between the energy cost and noise in neuronal response. In short, the CMFE sends a rich variety of information through broader communication bands (i.e., widely distributed firing rates) at the cost of accuracy. We demonstrate that the CMFE is reflected in the long-tailed, typically power law, distributions of inter-spike intervals obtained for the majority of recorded neurons. In other words, the power-law tails are more consistent with the CMFE rather than the MMI. Thus, we propose the mathematical principle by which cortical neurons may represent information about synaptic input into their output spike trains

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review