Networks of random trees as a model of neuronal connectivity

Evolutionary Advantages of Stimulus-Driven EEG Phase Transitions in the Upper Cortical Layers

Spatio-temporal brain activity monitored by EEG recordings in humans and other mammals has identified beta/gamma oscillations (20-80 Hz), which are self-organized into spatio-temporal structures recurring at theta/alpha rates (4-12 Hz). These structures have statistically significant correlations with sensory stimuli and reinforcement contingencies perceived by the subject. The repeated collapse of self-organized structures at theta/alpha rates generates laterally propagating phase gradients (phase cones), ignited at some specific location of the cortical sheet. Phase cones have been interpreted as neural signatures of transient perceptual experiences according to the cinematic theory of brain dynamics. The rapid expansion of essentially isotropic phase cones is consistent with the propagation of perceptual broadcasts postulated by Global Workspace Theory (GWT). What is the evolutionary advantage of brains operating with repeatedly collapsing dynamics? This question is answered using thermodynamic concepts. According to neuropercolation theory, waking brains are described as non-equilibrium thermodynamic systems operating at the edge of criticality, undergoing repeated phase transitions. This work analyzes the role of long-range axonal connections and metabolic processes in the regulation of critical brain dynamics. Historically, the near 10 Hz domain has been associated with conscious sensory integration, cortical "ignitions" linked to conscious visual perception, and conscious experiences. We can therefore combine a very large body of experimental evidence and theory, including graph theory, neuropercolation, and GWT. This cortical operating style may optimize a tradeoff between rapid adaptation to novelty vs. stable and widespread self-organization, therefore resulting in significant Darwinian benefits.

Primal-size neural circuits in meta-periodic interaction

Experimental observations of simultaneous activity in large cortical areas have seemed to justify a large network approach in early studies of neural information codes and memory capacity. This approach has overlooked, however, the segregated nature of cortical structure and functionality. Employing graph-theoretic results, we show that, given the estimated number of neurons in the human brain, there are only a few primal sizes that can be attributed to neural circuits under probabilistically sparse connectivity. The significance of this finding is that neural circuits of relatively small primal sizes in cyclic interaction, implied by inhibitory interneuron potentiation and excitatory inter-circuit potentiation, generate relatively long non-repetitious sequences of asynchronous primal-length periods. The meta-periodic nature of such circuit interaction translates into meta-periodic firing-rate dynamics, representing cortical information. It is finally shown that interacting neural circuits of primal sizes 7 or less exhaust most of the capacity of the human brain, with relatively little room to spare for circuits of larger primal sizes. This also appears to ratify experimental findings on the human working memory capacity.

Decay of connection probabilities with distance in 2D and 3D neuronal networks

We study connectivity in a model of a growing neuronal network in dimensions 2 and 3. Although the axon-to-dendrite proximity is an insufficient condition for establishing a functional synapse, it is still a necessary one. Therefore we study connection probabilities at short distances between the randomly grown axon trees and somas as probabilities of potential connections between the corresponding neurons. Our results show that, contrary to a common belief, these probabilities do not necessarily decay polynomially or exponentially in distance, but there are regimes of parameter values when the probability of proximity is not sensitive to the distance. In particular, in 3 dimensions the Euclidean distance between the neuronal cell body centers of neurons seems to play a very subtle role, as the probabilities of connections are practically constant within a certain finite range of distance. The model has a sufficient number of parameters to assess networks of neurons of different types. Our results give a firm basis for further modelling of the neuronal connectivity taking into account some realistic bouton distributions for establishing synaptic connections.