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Translational Neuroscience: How does bifurcation theory help us with epileptic surgery?

By Viktor Jirsa, Aix-Marseille University

Seizures  can occur spontaneously and  in a recurrent  manner,  which  defines  epilepsy;  or they  can be induced  in a normal  brain under  a variety  of conditions in most  neuronal  networks  and  species  from flies to humans. Such universality raises the possibility that invariant properties exist that characterize seizures under different physiological and pathological conditions. Starting from first principles of the theory of slow-fast systems in nonlinear dynamics, we conceptualize seizure dynamics mathematically and establish a taxonomy of seizures based on seizure onset and offset bifurcations. We demonstrate  that  only  five  state   variables  linked  by  integral-differential equations  are sufficient  to describe  the onset,  time course  and offset  of ictal-like  discharges as well as their recurrence.  These state variables define the model system called the Epileptor, where two state variables  are responsible for generating rapid  discharges (fast  time  scale),  two  for spike  and  wave  events  (intermediate time  scale)  and  one permittivity variable (slow  time  scale).  The permittivity variable captures effects evolving on slow timescales, including extracellular ionic concentrations and energy metabolism, with time delays of up to seconds as observed clinically. We propose  that normal  and  ictal  activities  coexist:  a separatrix acts  as  a barrier  (or seizure  threshold) between these  states. Seizure onset is reached upon the collision of normal brain trajectories with the separatrix. We show theoretically and experimentally how a system can be pushed toward seizure under a wide variety of conditions. Within our experimental model, the onset and offset of ictal-like discharges are well-defined mathematical events:  a saddle-node and homoclinic bifurcation, respectively. These bifurcations necessitate a baseline shift at onset and a logarithmic scaling of interspike intervals at offset.  These predictions were not only confirmed in our in vitro experiments, but also for focal seizures recorded in different syndromes, brain regions and species (humans and zebrafish). Extending this generic approach rooted in nonlinear dynamics towards human brain networks, we reconstruct personalized connectivity matrices of human epileptic patients using Diffusion Tensor weighted Imaging (DTI). Subsets of brain regions generating seizures in patients with refractory partial epilepsy are referred to as the epileptogenic zone (EZ). During a seizure, paroxysmal activity is not restricted to the EZ, but may recruit other brain regions and propagate activity through large brain networks, which comprise brain regions that are not necessarily epileptogenic. The identification of the EZ is crucial for candidates for neurosurgery and requires unambiguous criteria that evaluate the degree of epileptogenicity of brain regions. Stability analyses of propagating waves provide a set of indices quantifying the degree of epileptogenicity and predict conditions, under which seizures propagate to nonepileptogenic brain regions, explaining the responses to intracerebral electric stimulation in epileptogenic and nonepileptogenic areas. We demonstrate the predictive value of our seizure propagation model by validating it against empirical patient data.  In conjunction, our results provide guidance in the presurgical evaluation of epileptogenicity based on electrographic signatures in intracerebral electroencephalograms.