Epileptor Neural Fields

• Epileptor neural fields are spatially extended seizure models that promote local fast, intermediate, and slow dynamical systems to functions of space and time.
• They integrate short-range kernels and long-range connectivity to reproduce characteristic propagation speeds, bifurcation dynamics, and complex recruitment patterns observed in focal epilepsy.
• Patient-specific implementations leverage these models to predict seizure duration, inform stimulation protocols, and guide interventions through controlled perturbations and structural connectivity.

Epileptor neural fields are spatially extended seizure models in which each spatial location carries a local Epileptor or Epileptor-like dynamical system, and locations are coupled by short-range kernels, long-range connectivity, or both. In the formulation introduced for human focal epilepsy, the local state comprises a fast population $(u_1,u_2)$, an intermediate population $(q_1,q_2)$, and a slow permittivity variable $s$, so that seizure dynamics are represented as interactions across explicitly separated time scales embedded in a continuous spatial medium (Proix et al., 2017). Subsequent work has connected this framework to mechanistic neural masses with explicit SOM$^+$ and PV$^+$ interneuron populations (Ersöz et al., 2020), to general neural-field derivations and canonical continuum limits (Cook et al., 2021, Cooray et al., 2023), and to millimetre-scale patient-specific connectomes with delayed re-entry dynamics (Triebkorn et al., 6 Aug 2025).

1. Canonical formulation of the Epileptor field

The Epileptor neural field extends the local Epileptor by promoting all state variables to functions of space $x$ and time $t$. The field variables are

$$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$

where $(u_1,u_2)$ form the fast population associated with low-voltage fast activity, $(q_1,q_2)$ form the intermediate population associated with spike-and-wave discharges, and $(q_1,q_2)$0 is the slow permittivity variable that controls transitions between interictal and ictal states. The key local excitability parameter is $(q_1,q_2)$1; for $(q_1,q_2)$2 the local system can autonomously generate seizures (Proix et al., 2017).

In its neural-field form, the model is an integral field with both local homogeneous and long-range heterogeneous couplings. The fast subsystem obeys

$(q_1,q_2)$3

$(q_1,q_2)$4

and the slow variable satisfies

$(q_1,q_2)$5

The intermediate population obeys

$(q_1,q_2)$6

$(q_1,q_2)$7

Here $(q_1,q_2)$8 is a Heaviside firing-rate nonlinearity, the short-range kernels are spatial convolutions

$(q_1,q_2)$9

and the canonical local kernel is Laplacian-like,

$s$0

The term $s$1 represents tractography-derived long-range connectivity between distinct fields (Proix et al., 2017).

This construction gives a continuous field of local 5D Epileptors. Each location carries two oscillatory subpopulations and one slow control variable, while short-range coupling captures local recruitment and long-range coupling captures network-level spread. Spatial heterogeneity enters principally through $s$2, which can be set higher in epileptogenic tissue and lower at the boundaries to prevent recruitment (Proix et al., 2017).

2. Local node architecture and multiscale organization

The defining feature of Epileptor neural fields is not only spatial coupling but also explicit separation of time scales at the node level. In the canonical Epileptor field, these scales are fast $s$3, intermediate $s$4, and slow $s$5, with an additional ultra-slow contribution entering through the temporal convolution $s$6 (Proix et al., 2017).

A mechanistic realization of this architecture appears in the Wendling-type neural mass extended with pyramidal cells, SOM$s$7 interneurons, and PV$s$8 interneurons. In that model the state variables are the mean PSPs $s$9, the pulse-to-wave transfer is

$^+$0

and the linear synaptic response is

$^+$1

After reduction and rescaling, the system becomes an explicit three-time-scale slow-fast system with time-scale ratios

$^+$2

and three separated subsystems: fast $^+$3, slow $^+$4, and super-slow $^+$5. With the parameters $^+$6 s, $^+$7 s, and $^+$8 s, the SOM$^+$9 population provides the slowest dynamics and functions as a permissivity variable that shifts the excitability of the remaining subsystem (Ersöz et al., 2020).

The same paper states that the Epileptor is not mentioned explicitly, but the model is structurally Epileptor-like. Bursting is classified as fold/Hopf bursting in the sense of Izhikevich: during the quiescent phase the trajectory follows the stable lower branch of a superslow manifold, crosses a fold, jumps to a spiking branch, and later returns after crossing a second fold. In this interpretation the super-slow SOM$^+$0 variables $^+$1 act as mechanistic permissivity variables, while the PV$^+$2-pyramidal subsystem generates the fast oscillatory component (Ersöz et al., 2020).

A reduced 2D Epileptor is also used as the local unit in a patient-specific millimetre-scale field. There the fast variable $^+$3 and slow variable $^+$4 satisfy

$^+$5

$^+$6

with $^+$7 a Heaviside function and $^+$8. This version retains the slow-fast seizure logic while embedding it directly in delayed structural coupling (Triebkorn et al., 6 Aug 2025).

3. Spatial coupling, wavefronts, and propagation regimes

A central result of the Epileptor field is that seizure spread is not described by a single wave mechanism. The model distinguishes a slow ictal wavefront in the fast population from rapid spike-and-wave propagation in the intermediate population. When a seizure starts locally, neighboring sites are recruited sequentially by the short-range term $^+$9, and the resulting traveling front links the interictal fixed point to the ictal limit cycle. Numerical shooting in a reduced fast subsystem yields a front solution $x$0, and the resulting speed $x$1 is slow, matching approximately $x$2 mm/s experimentally. By contrast, spike-and-wave discharges propagate as a phase wave in a chain of coupled oscillators, with propagation speed $x$3 approximately two orders of magnitude larger than $x$4, consistent with experimental SWD velocities of $x$5–$x$6 mm/s (Proix et al., 2017).

The two propagation modes arise from different dynamical structures. The slow ictal wavefront is an excitable-media front in the fast population, whereas SWD propagation is a coupled-oscillator phenomenon in the intermediate population. The model further shows that low-voltage fast activity hampers the slow front: when $x$7 oscillates above and below the Heaviside threshold, the effective coupling is intermittently switched on and off, producing fronts that are an order of magnitude slower than in a reduction that eliminates the fast oscillatory component (Proix et al., 2017).

The same framework explains why the source of SWDs can be stationary or moving. An effective slow drive $x$8, determined by the slow permittivity and time-averaged fast activity, governs local oscillator frequency in the $x$9-population. If epileptogenicity is similar across the recruited region, the maximum of $t$0 can move with the seizure, and the SWD source often tracks the ictal wavefront. If the onset region is more epileptogenic, its $t$1 remains maximal and the SWD source stays stationary at onset (Proix et al., 2017).

Patient-specific implementations add anatomically explicit local and global propagation. In a millimetre-scale virtual brain, each vertex is coupled by delayed white-matter connectivity $t$2 with tract lengths $t$3 and conduction speed $t$4, and by local geodesic or Euclidean coupling $t$5 with $t$6. The resulting field spans cortex, hippocampus, cerebellum, and subcortical grids, with approximately $t$7 vertices, and produces traveling waves, white-matter jumps, spiral waves, and ring-like surface waves (Triebkorn et al., 6 Aug 2025).

General neural-field theory provides the derivational background for such constructions. A unifying framework expresses fields through temporal kernels $t$8, spatial kernels $t$9, and transfer functions $$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$0, while a continuum limit of coupled neural masses yields field equations of the form

$$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$1

and, under finite conduction velocity, damped wave operators of Robinson type (Cook et al., 2021). A separate continuum derivation shows that when neural masses are placed on a 2D lattice and linearized near a stable fixed point, the field limit becomes a set of coupled real Klein-Gordon equations; the same paper explicitly suggests treating the Epileptor as the local neural mass at each lattice node when constructing such fields (Cooray et al., 2023).

4. Onset, offset, and transient epileptiform dynamics

In the Epileptor field, seizure onset is governed by bifurcation structure within the fast subsystem as slow variables drift. Local onset can be triggered either by slow drift in $$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$2 due to $$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$3 or by external stimulus, after which local coupling recruits neighboring sites. The intermediate population $$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$4 enters its oscillatory regime through a SNIC bifurcation as the effective parameter $$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$5 crosses a critical value, which is why SWDs emerge only during a specific seizure stage. Seizure termination in the fast population occurs through a homoclinic bifurcation; near offset the trajectory approaches a separatrix, and a propagating SWD can push multiple sites across it nearly simultaneously, producing quasi-synchronous termination within a connected cluster (Proix et al., 2017).

The field also predicts asynchronous clustered termination. In a two-field configuration with heterogeneous long-range coupling, strong $$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$6 permits SWDs to propagate effectively between fields and favors quasi-synchronous termination across both. Weak $$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$7 prevents reliable SWD transfer, so each field behaves as an independent ictal cluster with quasi-synchronous termination internally but large delays between clusters. In simulations, large termination delays occur only for low connection strength; in human SEEG, pairs with large termination delays have significantly weaker structural connectivity, and clusters with larger delays differ significantly in SWD correlation and tractography weight with Mann–Whitney $$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$8 (Proix et al., 2017).

A distinct but related mechanism for epileptiform transients is provided by fold of cycles dynamics in a thalamo-cortical slow-fast model. There, a stable fixed point and a large-amplitude oscillation coexist after a fold of cycles bifurcation, and just before that bifurcation the system is excitable: a suprathreshold pulse can push the trajectory into the ghost of the forthcoming stable and saddle cycles, generating a prolonged rhythmic transient before deterministic return to baseline. This mechanism yields self-terminating spike-wave after-discharges without ultraslow parameter drift and has been proposed as complementary to classical Epileptor-style slow-drift seizure mechanisms (Baier et al., 2016).

The mechanistic Wendling-SOM/PV mass provides a further onset scenario. Decreasing the SOM$$u_1(x,t),\ u_2(x,t),\ q_1(x,t),\ q_2(x,t),\ s(x,t),$$9 synaptic gain $(u_1,u_2)$0 produces a Z-shaped equilibrium curve, sporadic bursts in a bistable region, sustained pre-ictal bursting, and eventually low-voltage fast onset gamma activity around $(u_1,u_2)$1 Hz through Hopf bifurcations $(u_1,u_2)$2 and $(u_1,u_2)$3. In that model the super-slow SOM$(u_1,u_2)$4 subsystem controls fold/Hopf bursting, while increasing PV$(u_1,u_2)$5 gain $(u_1,u_2)$6 or excitatory input $(u_1,u_2)$7 shifts the system toward low-voltage fast onset (Ersöz et al., 2020).

5. Control, stimulation, and structural intervention

Because the local dynamics are slow-fast, control strategies in Epileptor neural fields are naturally phrased as perturbations that move the system away from seizure-supporting manifolds or away from delay-supported re-entrant loops. In the Wendling-SOM/PV model, stimulation is introduced additively into the sigmoidal input of each population, and the authors analyze both direct current shifts and biphasic pulses. The pulses are charge-balanced, with $(u_1,u_2)$8 ms per phase and total duration $(u_1,u_2)$9 ms, and frequencies ranging from low values to $(q_1,q_2)$0 Hz. The key reported result is that intermediate stimulation frequencies $(q_1,q_2)$1 Hz can abort seizures if the timescale difference is pronounced. Geometrically, positive SOM$(q_1,q_2)$2-only stimulation shifts the SOM$(q_1,q_2)$3 nullsurface so that a stable equilibrium lies on the lower branch of the slow manifold, trapping the trajectory and eliminating bursting; hyperpolarizing pyramidal cells can also suppress bursting but is less efficient; depolarizing PV$(q_1,q_2)$4 interneurons is inferred to favor bursting (Ersöz et al., 2020).

The same study reports a strong dependence on synaptic time constants. Because SOM$(q_1,q_2)$5 interneurons have the largest time constant $(q_1,q_2)$6, relatively low stimulation frequencies can act as an effective direct-current bias on the SOM$(q_1,q_2)$7 subsystem while having weaker effects on faster populations. In simulations, SOM$(q_1,q_2)$8-only stimulation with $(q_1,q_2)$9 can abort bursting with lower frequency and amplitude, for example $(q_1,q_2)$00 Hz and amplitude $(q_1,q_2)$01, whereas homogeneous stimulation of all populations with $(q_1,q_2)$02 requires a higher frequency, for example $(q_1,q_2)$03 Hz, for full suppression (Ersöz et al., 2020).

In patient-specific delayed Epileptor fields, control is phase dependent. Re-entry is terminated by precisely timed charge-balanced biphasic pulses delivered through a single SEEG-like contact in anterior temporal lobe. The cathodic phase is $(q_1,q_2)$04s, the anodic phase is $(q_1,q_2)$05s at half amplitude, and three pulses are separated by one inter-spike interval each. When the stimulation onset phase is varied across the inter-spike interval, pulses terminate re-entry only in narrow phase windows; outside those windows the effect is absent or termination is delayed by a few cycles. The abstract states that precisely timed biphasic stimuli abort re-entry in silico and yield phase-dependent termination rules validated in intracranial recordings (Triebkorn et al., 6 Aug 2025).

Structural disconnection is modeled in the same framework. Radio-frequency thermocoagulation is implemented as spherical lesions of radius $(q_1,q_2)$06 mm around eight contacts in anterior temporal white matter, with all intersecting streamlines removed. Multiple subpial transection is implemented as parallel cuts across gyri in the epileptogenic zone every $(q_1,q_2)$07 mm, removing local edges shorter than $(q_1,q_2)$08 mm. Both manipulations can either abolish or unmask re-entry, depending on where the operating point lies in the delay-coupling corridor. RFTC disproportionately removes long corticocortical fibers and can either weaken loops or lengthen effective path delays; MST primarily reduces local recurrence and shifts the upper coupling boundary for re-entry downward (Triebkorn et al., 6 Aug 2025).

6. Patient-specific implementations and broader theoretical context

The most detailed realization of an Epileptor neural field to date embeds reduced 2D Epileptors in a drug-resistant epilepsy patient’s structural connectome reconstructed from $(q_1,q_2)$09 mm T1w MRI and $(q_1,q_2)$10 mm isotropic diffusion MRI with $(q_1,q_2)$11 diffusion directions. Vertex volumes are estimated for cortical surfaces and matched in subcortical grids to $(q_1,q_2)$12, local coupling is computed from geodesic or Euclidean distances, and white-matter connectivity weights are derived from $(q_1,q_2)$13 million tractography streamlines with SIFT2 weighting. Within this system, re-entry emerges only in a narrow delay-coupling corridor: on an anterior temporal patch with $(q_1,q_2)$14, re-entry appears for $(q_1,q_2)$15; below that propagation fails, and above that the entire patch is recruited too quickly and becomes globally refractory (Triebkorn et al., 6 Aug 2025).

The same work introduces activity-dependent slowing by replacing the fixed intrinsic time scale $(q_1,q_2)$16 with $(q_1,q_2)$17 and evolving $(q_1,q_2)$18 toward $(q_1,q_2)$19 during the active phase and toward $(q_1,q_2)$20 during the refractory phase, with $(q_1,q_2)$21, $(q_1,q_2)$22, and $(q_1,q_2)$23. This modification yields seizure frequency slowing toward termination and supports near-synchronous offset in extended epileptogenic and propagation zones (Triebkorn et al., 6 Aug 2025).

A principal empirical claim of that study is that the delay-constrained re-entry window predicts frequency and seizure duration across $(q_1,q_2)$24 recorded seizures. In the clinical data, a small focal seizure involving three closely spaced electrodes lasted $(q_1,q_2)$25 s with $(q_1,q_2)$26 Hz offset, whereas a larger widespread seizure involving eleven broadly spaced electrodes lasted $(q_1,q_2)$27 s with $(q_1,q_2)$28 Hz offset. Across all seizures, offset frequency versus mean contact distance showed $(q_1,q_2)$29, and seizure duration versus mean contact distance showed $(q_1,q_2)$30, matching the model prediction that larger loops support lower offset frequency and longer duration (Triebkorn et al., 6 Aug 2025).

Beyond focal epilepsy, neural-field approaches have been generalized to corticothalamic-basal ganglia systems. In that setting, oscillations arise as loop resonances in corticothalamic and cortico-basal-ganglia-thalamic circuits, and the same stability surface can support absence-like $(q_1,q_2)$31 Hz activity, tonic-clonic $(q_1,q_2)$32 Hz activity, and Parkinsonian $(q_1,q_2)$33 Hz and $(q_1,q_2)$34 Hz rhythms. This is not an Epileptor field in the strict local-state-variable sense, but it places epilepsy and other pathological oscillations within a shared neural-field framework of loop gains, delays, and bifurcation boundaries (Müller et al., 2024).

At the most general level, neural field theory provides the mathematical envelope in which Epileptor neural fields sit. Unified derivations begin from coarse-grained population variables on a continuous cortical sheet, use temporal kernels $(q_1,q_2)$35, spatial kernels $(q_1,q_2)$36, and transfer functions $(q_1,q_2)$37, and recover first-order Amari/Wilson–Cowan equations, second-order PSP-based models, and finite-velocity damped wave equations as special cases (Cook et al., 2021). A separate continuum-limit argument shows that when topologically equivalent local neural masses are placed on a 2D lattice and linearized near a stable fixed point, the resulting field theory is invariant across those local choices and reduces to coupled real Klein-Gordon fields. The same source explicitly proposes reading the Epileptor as the local neural mass at each lattice node, which suggests a canonical small-amplitude field description for the fast component of Epileptor-based spatial models (Cooray et al., 2023).

Epileptor neural fields therefore occupy a spectrum of formulations. At one end are phenomenological integral fields of local 5D Epileptors; at another are mechanistic neural masses in which SOM$(q_1,q_2)$38, PV$(q_1,q_2)$39, thalamic, or corticothalamic populations realize the slow and fast subsystems explicitly; and at the largest scale are patient-specific delayed fields in which seizure maintenance can be governed by re-entry. Across these variants, the common structure is a spatially coupled multiscale system in which seizure onset, spread, and termination are controlled by the interaction of local bifurcation geometry with anatomical connectivity and propagation delay (Proix et al., 2017).