The interaction between neural populations: Additive versus diffusive coupling

Models of networks of populations of neurons commonly assume that the interactions between neural populations are via additive or diffusive coupling. When using the additive coupling, a population's activity is affected by the sum of the activities of neighbouring populations. In contrast, when using the diffusive coupling a neural population is affected by the sum of the differences between its activity and the activity of its neighbours. These two coupling functions have been used interchangeably for similar applications. Here, we show that the choice of coupling can lead to strikingly different brain network dynamics. We focus on a model of seizure transitions that has been used both with additive and diffusive coupling in the literature. We consider networks with two and three nodes, and large random and scale-free networks with 64 nodes. We further assess functional networks inferred from magnetoencephalography (MEG) from people with epilepsy and healthy controls. To characterize the seizure dynamics on these networks, we use the escape time, the brain network ictogenicity (BNI) and the node ictogenicity (NI), which are measures of the network's global and local ability to generate seizures. Our main result is that the level of ictogenicity of a network is strongly dependent on the coupling function. We find that people with epilepsy have higher additive BNI than controls, as hypothesized, while the diffusive BNI provides the opposite result. Moreover, individual nodes that are more likely to drive seizures with one type of coupling are more likely to prevent seizures with the other coupling function. Our results on the MEG networks and evidence from the literature suggest that the additive coupling may be a better modelling choice than the diffusive coupling, at least for BNI and NI studies. Thus, we highlight the need to motivate and validate the choice of coupling in future studies.

Amplitude Death Induced by Intrinsic Noise in a System of Three Coupled Stochastic Brusselators

In this work, we study the interplay between intrinsic biochemical noise and the diffusive coupling, in an array of three stochastic Brusselators that present a limit-cycle dynamics. The stochastic dynamics is simulated by means of the Gillespie algorithm. The intensity of the intrinsic biochemical noise is regulated by changing the value of the system volume (Ω), while keeping constant the chemical species' concentration. To characterize the system behavior, we measure the average spike amplitude (ASA), the order parameter R, the average interspike interval (ISI), and the coefficient of variation (CV) for the interspike interval. By analyzing how these measures depend on Ω and the coupling strength, we observe that when the coupling parameter is different from zero, increasing the level of intrinsic noise beyond a given threshold suddenly drives the spike amplitude, SA, to zero and makes ISI increase exponentially. These results provide numerical evidence that amplitude death (AD) takes place via a homoclinic bifurcation.

Exotic Patterns of Synchrony in Planar Lattice Networks

Patterns of dynamical synchrony that can occur robustly in networks of coupled dynamical systems are associated with balanced colorings of the nodes of the network. In symmetric networks, the orbits of any group of symmetries automatically determine a balanced orbit coloring. Balanced colorings not of this kind are said to be exotic. Exotic colorings occur in infinite planar lattices, both square and hexagonal, with various short-range couplings. In some cases, a balanced two-coloring remains balanced when colors are swapped along suitable diagonals, giving rise to uncountably many distinct exotic colorings. We explain this phenomenon in terms of iterated orbit colorings, in which the quotient of the lattice by an orbit coloring has extra symmetries, allowing new orbit colorings on the quotient, which then lift back to the lattice. We apply the same construction to several other exotic lattice colorings. Two appendices discuss how to modify the notion of balance for networks with diffusive coupling, and how to formalize the differential equations in infinitely many variables that arise for lattices.

A Barter Economy in Tumors: Exchanging Metabolites through Gap Junctions

To produce physiological functions, many tissues require their cells to be connected by gap junctions. Such diffusive coupling is important in establishing a cytoplasmic syncytium through which cells can exchange signals, substrates and metabolites. Often the benefits of connectivity become apparent solely at the multicellular level, leading to the notion that cells work for a common good rather than exclusively in their self-interest. In some tumors, gap junctional connectivity between cancer cells is reduced or absent, but there are notable cases where it persists or re-emerges in late-stage disease. Diffusive coupling will blur certain phenotypic differences between cells, which may seem to go against the establishment of population heterogeneity, a central pillar of cancer that stems from genetic instability. Here, building on our previous measurements of gap junctional coupling between cancer cells, we use a computational model to simulate the role of connexin-assembled channels in exchanging lactate and bicarbonate ions down their diffusion gradients. Based on the results of these simulations, we propose that an overriding benefit of gap junctional connectivity may relate to lactate/bicarbonate exchange, which would support an elevated metabolic rate in hypoxic tumors. In this example of barter, hypoxic cancer cells provide normoxic neighbors with lactate for mitochondrial oxidation; in exchange, bicarbonate ions, which are more plentiful in normoxic cells, are supplied to hypoxic neighbors to neutralize the H+ ions co-produced glycolytically. Both cells benefit, and so does the tumor.

Dynamical modes of two almost identical chemical oscillators connected via both pulsatile and diffusive coupling

The dynamics of two almost identical chemical oscillators with mixed diffusive and pulsatile coupling are systematically studied.