Draft:Epileptor-2

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Epileptor-2 is one of the models of neural computation that describes in a simple manner the epileptic discharges.^[1] observed in the brain,^[2] named after the previous model called Epileptor.^[3] It replicates both brief interictal discharges (Spike-and-wave) and and longer ictal discharges constituting seizures. In the model, the interictal discharges are observed as clusters of action potentials in the activity of individual neurons, and the ictal discharges are represented as clusters of those shorter discharges.

According to the Epileptor-2 model, brief interictal discharges are characterized as stochastic oscillations of the membrane potential and synaptic resource, while ictal discharges emerge as oscillations in the extracellular concentration of potassium ions and the intracellular concentration of sodium ions. The both models, Epileptor and Epileptor-2, demonstrate that ionic dynamics play a decisive role in the generation of pathological activity. The model has been applied for simulation of the effects of electrical stimulation on epileptic activity in vitro^[4]

Below, a modified Epileptor-2 model is described, where the second extracellular compartment has been added. The model is governed by the system of the 5th order system of ordinary differential equations. Ionic dynamics is considered for two extracellular concentrations of potassium ions, at the proximity of neurons, [ K ] o 1 {\displaystyle [K]_{o1}} , and at a distance, [ K ] o 2 {\displaystyle [K]_{o2}} , and the intracellular concentration of sodium ions, [ N a ] i {\displaystyle [Na]_{i}} :

d [ K ] o 1 d t = [ K ] o 2 − [ K ] o 1 τ K 1 + δ K   ν ( t ) − 2 β I p u m p ( t ) {\displaystyle {d[K]_{o1} \over dt}={[K]_{o2}-[K]_{o1} \over \tau _{K1}}+\delta _{K}~\nu (t)-2\beta I_{pump}(t)}

d [ K ] o 2 d t = [ K ] b a t h − [ K ] o 2 τ K 2 + [ K ] o 1 − [ K ] o 2 τ K 1 {\displaystyle {d[K]_{o2} \over dt}={[K]_{bath}-[K]_{o2} \over \tau _{K2}}+{[K]_{o1}-[K]_{o2} \over \tau _{K1}}}

d [ N a ] i d t = [ N a ] i 0 − [ N a ] i τ N a + δ N a ν ( t ) − 3 I p u m p ( t ) {\displaystyle {d[Na]_{i} \over dt}={{[Na]_{i}}^{0}-[Na]_{i} \over \tau _{Na}}+\delta _{Na}\nu (t)-3I_{pump}(t)}

According to these equations, the concentrations increase because of firing with the rate ν {\displaystyle \nu } , relax with the time scales τ K 1 ,   τ K 2 {\displaystyle \tau _{K1},~\tau _{K2}} , and τ N a {\displaystyle \tau _{Na}} , and decrease due to the sodium-potassium pump I p u m p {\displaystyle I_{pump}} .

The membrane polarization V {\displaystyle V} averaged across population and the synaptic resource x {\displaystyle x} are governed by

d V d t = u ( t ) − V τ m {\displaystyle {dV \over dt}={u\left(t\right)-V \over \tau _{m}}}

d x d t = 1 − x τ x − δ x x   ν ( t ) {\displaystyle {dx \over dt}={1-x \over \tau _{x}}-\delta _{x}x~\nu (t)}

where

u ( t ) = g K ( V K ( t ) − V K 0 ) + G s y n ( x − 0.5 )   ν ( t ) + σ   η ( t ) {\displaystyle u(t)=g_{K}(V_{K}(t)-V_{K}^{0})+G_{syn}(x-0.5)~\nu (t)+\sigma ~\eta (t)}

The source term u ( t ) {\displaystyle u(t)} summarises the current mediated by the changes ot the potassium concentration, the synaptic current and the noise η ( t ) {\displaystyle \eta (t)} (white gaussian noise). The synaptic resource decreases because of firing with the rate ν {\displaystyle \nu } and recovers with the time scale τ x {\displaystyle \tau _{x}} .

The potassium reversal potential is

V K ( t ) = 26.6   ln ⁡ ( [ K ] o 2 / 130 ) {\displaystyle V_{K}(t)=26.6~\ln([K]_{o2}/130)}

The sodium–potassium pump is

I p u m p ( t ) = ρ / ( 1 + exp ⁡ ( 3.5 − [ K ] o 1 ) / ( 1 + exp ⁡ ( ( 25 − [ N a ] o 1 ) / 3 ) ) {\displaystyle I_{pump}(t)=\rho /\left(1+\exp(3.5-[K]_{o1})/(1+\exp((25-[Na]_{o1})/3)\right)}

The sigmoidal dependence of the firing rate on voltage is

ν ( t ) = 100 ( 2 / ( 1 + exp ⁡ ( ( 6 − V ) / 10 ) ) − 1 ) {\displaystyle \nu (t)=100\left(2/\left(1+\exp((6-V)/10)\right)-1\right)}

Representative neuron is governed by an additional quadratic integrate and fire model for the membrane potential U {\displaystyle U} . It receives the same input u ( t ) {\displaystyle u(t)} as the population:

d U d t = g U C U ( U − U 1 ) ( U − U 2 ) + g L C U u ( t ) ,           i f       U > U t h     t h e n     U = U r e s e t {\displaystyle {dU \over dt}={g_{U} \over C_{U}}(U-U_{1})(U-U_{2})+{g_{L} \over C_{U}}u(t),~~~~~if~~~U>U^{th}~~then~~U=U_{reset}}

Figure below illustrates experimentally registered and simulated epileptic activity. In the experiment, two ictal discharges were recorded simultaneously as the membrane potential in a single neuron and the extracellular potassium concentration. Similar traces were obtained in the model, which also predicts the intracellular sodium concentration, the averaged across population depolarization of neurons, and the change of synaptic resource. The parameters used for simulations in Figure are as follows:

τ K 1 = 25 s ,   τ K 2 = 250 s ,   τ N a = 20 s ,   τ x = 2 s ,   τ m = 0.01 s , {\displaystyle \tau _{K1}=25s,~\tau _{K2}=250s,~\tau _{Na}=20s,~\tau _{x}=2s,~\tau _{m}=0.01s,}   δ K = 0.04   m M ,   δ N a = 0.03   m M ,   δ x = 0.01 ,   ρ = 0.8 ,   β = 10 ,   σ = 8 , {\displaystyle ~\delta _{K}=0.04~mM,~\delta _{Na}=0.03~mM,~\delta _{x}=0.01,~\rho =0.8,~\beta =10,~\sigma =8,}   G s y n / g L = 2.5   m V ⋅ s ,   g K / g L = 0.5 ,   g L = 5   n S , {\displaystyle ~G_{syn}/g_{L}=2.5~mV\cdot s,~g_{K}/g_{L}=0.5,~g_{L}=5~nS,} [ K ] o 0 = 3   m M ,   [ K ] b a t h = 8.5   m M ,   [ N a ] i 0 = 10   m M , {\displaystyle [K]_{o}^{0}=3~mM,~[K]_{bath}=8.5~mM,~[Na]_{i}^{0}=10~mM,} g U = 0.4 ,   C U = 0.2 ,   U t h = 25 ,   U r e s e t = − 50 ,   U 1 = − 60 ,   U 2 = − 40 ,   U 0 = − 70 {\displaystyle g_{U}=0.4,~C_{U}=0.2,~U^{th}=25,~U_{reset}=-50,~U_{1}=-60,~U_{2}=-40,~U_{0}=-70}

The original model's implementation.^[5] in Python (programming language) is available from the online database ModelDB^[6]

References