On the modelling of seizure dynamics

In this issue of Brain, Jirsa and colleagues offer a masterful account of epilepsy that discloses the universal and invariant properties of seizure dynamics (Jirsa et al., 2014). They derive a formal taxonomy of seizure activity from the basic principles of coupled dynamical systems. In brief, they show that a minimal model— comprising just five states or variables—is sufficient to describe the onset, time-course and offset of seizure activity. This may sound implausible; however, the repertoire of dynamics that coupled systems can generate may be much smaller than people imagine. In effect, Jirsa et al. use their understanding of dynamical systems to ‘diagnose’ the mathematical aetiology of the ‘signs and symptoms’ expressed in the statistics of seizure activity. Put simply, the existence of dissociable fast discharges and spike-wave events immediately tells them that the dynamics must be generated by two pairs of (hidden) coupled states generating fast and spike-wave oscillations, respectively. The fact that seizure activity waxes and wanes over a slower timescale then calls for a further slow permittivity variable that controls the expression of fast dynamics. Within this model, the onset and offset of seizure activity become well-defined mathematical objects: a saddle node and homoclinic bifurcation, respectively. The tell-tale signatures of these bifurcations were subsequently confirmed using in vitro experiments and focal seizures recorded in humans and zebrafish. So why is this important? Without recourse to Wikipedia, I confess that I could not explain the difference between a saddle node and homoclinic bifurcation. However, this is not important: the authors have done all the mathematical heavy lifting for us and bring three things to the table: first, the notion that seizure onset (and offset) can be cast in terms of bifurcations (qualitative changes in dynamics) that are induced by changing the parameters of a coupled dynamical system—such as synaptic efficacy in neuronal microcircuits. Second, the attending qualitative changes rest upon a separation of temporal scales. For example, slow fluctuations in synaptic efficacy induce qualitative changes in fast dynamics. Finally, there is an inherent circular causality in epileptogenesis; whereby slow fluctuations in synaptic efficacy cause changes in fast neuronal activity—that couple back to the slow variables through processes such as activity-dependent plasticity. This circular causality is a necessary aspect of coupling variables that fluctuate over separable timescales. It is interesting to note that these fundaments of dynamical systems are formally articulated in synergetics—an approach to self-organization pioneered by the first author’s mentor Herman Haken (Haken, 1983). In short, what is on offer here is a canonical description of seizure activity that has all the necessary ingredients for a formal taxonomy of seizure dynamics. But how does a mathematical taxonomy reach beyond phenomenology? At this point, we have to consider why the modelling of seizure dynamics is potentially important. In previous years, there have been substantial advances in the use of models of neuronal microcircuits as forward or generative models of observed electrophysiological responses. This is most evident in the advent of dynamic causal modelling, which is now used routinely to address questions about functional brain architectures—and their synaptic underpinnings—in cognitive and systems neuroscience. The key thing here is that dynamic causal models are not just used to reproduce the phenomenology of electrophysiological measurements: they explain empirical data through Bayesian model inversion and subsequent selection. A formal understanding of how seizure dynamics are generated could therefore provide vital constraints on dynamic causal models of seizure activity. In principle, this means that one could identify the biophysical substrates of slow and fast variables in phenomenological models.