pubmed.ncbi.nlm.nih.gov

Minimal model of interictal and ictal discharges "Epileptor-2" - PubMed

  • ️Mon Jan 01 2018

Minimal model of interictal and ictal discharges "Epileptor-2"

Anton V Chizhov et al. PLoS Comput Biol. 2018.

Erratum in

Abstract

Seizures occur in a recurrent manner with intermittent states of interictal and ictal discharges (IIDs and IDs). The transitions to and from IDs are determined by a set of processes, including synaptic interaction and ionic dynamics. Although mathematical models of separate types of epileptic discharges have been developed, modeling the transitions between states remains a challenge. A simple generic mathematical model of seizure dynamics (Epileptor) has recently been proposed by Jirsa et al. (2014); however, it is formulated in terms of abstract variables. In this paper, a minimal population-type model of IIDs and IDs is proposed that is as simple to use as the Epileptor, but the suggested model attributes physical meaning to the variables. The model is expressed in ordinary differential equations for extracellular potassium and intracellular sodium concentrations, membrane potential, and short-term synaptic depression variables. A quadratic integrate-and-fire model driven by the population input current is used to reproduce spike trains in a representative neuron. In simulations, potassium accumulation governs the transition from the silent state to the state of an ID. Each ID is composed of clustered IID-like events. The sodium accumulates during discharge and activates the sodium-potassium pump, which terminates the ID by restoring the potassium gradient and thus polarizing the neuronal membranes. The whole-cell and cell-attached recordings of a 4-AP-based in vitro model of epilepsy confirmed the primary model assumptions and predictions. The mathematical analysis revealed that the IID-like events are large-amplitude stochastic oscillations, which in the case of ID generation are controlled by slow oscillations of ionic concentrations. The IDs originate in the conditions of elevated potassium concentrations in a bath solution via a saddle-node-on-invariant-circle-like bifurcation for a non-smooth dynamical system. By providing a minimal biophysical description of ionic dynamics and network interactions, the model may serve as a hierarchical base from a simple to more complex modeling of seizures.

PubMed Disclaimer

Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. Schematic of Epileptor-2 and the described mechanisms of interictal and ictal discharges.

(A) The model consists of the equations that describes the ionic dynamics, the neuronal population excitation and the activity of a single representative neuron driven by the population. (B) Mechanisms of the discharges, explained in the main text.

Fig 2
Fig 2. Firing rate-versus-depolarization dependence, Eq 5.
Fig 3
Fig 3. Simulation of an interictal regime with spontaneous short bursts (SBs).

(A) The population firing rate. (B) The nominal depolarization. (C) The synaptic resource. (D) The intracellular sodium (red line) and extracellular potassium (blue) concentrations. (E) The membrane voltage in a representative neuron. (F) The membrane voltage on the time interval containing four SBs and marked by the dashed line in E. (G) The ionic flux through the Na-K-pump (orange, right axis) and the ionic concentrations (left axis) zoomed from D. Population variables in A-D and G were calculated with Eqs 1–8 with the basic parameter set except one different value, τK = 10s. The neuron membrane potential in E, F was obtained from Eqs 9 and 10. Note a lack of a strong clustering of SBs.

Fig 4
Fig 4. Simulations of IDs, each consisting of separate short bursts.

(A) The nominal depolarization (top plot), the intracellular sodium (red line) and extracellular potassium (blue) concentrations (bottom, left axis), and the ionic flux through the Na-K-pump (orange line, bottom plot, right axis) during six IDs. (B) The population firing rate (black), the nominal depolarization (red), and the synaptic resource (violet) during a single ID consisting of some SBs. (C) Zoomed traces from A, bottom during a single ID. Simulations were done with Eqs 1–8 with the basic parameter set. Note the decrease of [K]o at the high level of peaks of Ipump and following the termination of the ID.

Fig 5
Fig 5. Simulations of a single neuron activity during the IDs shown in Fig 4.

(A) Two IDs as bursts of clustered SBs, seen in the membrane voltage. (B) A single ID containing a number of SBs. Membrane voltage. (C) A single SB.

Fig 6
Fig 6. Ictal discharges recorded in a pyramidal neuron from a rat entorhinal cortex using an in vitro 4-AP model of epileptic activity.

(A) The full record. (B, C) Zoomed fragments. Note each ID is a burst of SBs of spikes.

Fig 7
Fig 7. Ion concentration changes during seizures.

Modified with permission from [15].

Fig 8
Fig 8. Proportionality between the excitatory and inhibitory currents and the firing rate in the experiment.

(A) Upper trace: A representative recording of glutamatergic synaptic activity in the voltage-clamp mode at the reversal potential of GABA-mediated currents (Vhold = VGABA = −51mV). Lower trace: A corresponding firing activity of the neighboring neuron recorded in the cell-attached voltage-clamp mode. (B) The recordings from the same pair of neurons as in A. The upper trace was recorded at the reversal potential of glutamate-mediated currents (Vhold = VGlut = 0) and represents the GABAergic synaptic activity. The yellow shadow indicates the IDs. (C) Several superimposed IID-like events from A (left) and B (right) that correspond to SBs and constitute IDs. (D) Pooled data from four IDs, each consisting of 50–80 SBs as in C. A histogram represents the firing rate of the neuron recorded in the cell-attached VC mode. Solid lines are the plots of GABA- and glutamate-mediated currents scaled to represent the synaptic input at the membrane potential of -40 mV.

Fig 9
Fig 9. Fast subsystem producing SBs.

(A) The phase space of the system of Eqs 3’–5’. Nullclines (solid red lines) and vector-fields (arrows) are calculated for [K]O=[K]O0; dashed nullclines are for [K]O=[K]O0+10mM. Purple curve is a trajectory. (B) the traces of V(t) and xD(t). The parameter values correspond to the basic set of parameters, except the noise amplitude σ/gL = 40 mV.

Fig 10
Fig 10. Slow subsystem producing IDs.

(A) IDs in the full Epileptor-2 model. (B) Oscillations in the reduced model based on Eqs 1’, 2’, 5” and 8’. The plots repeat those in Fig 4. (C, D, and E) Phase portraits of the reduced model obtained for [K]bath = 3, 6.42017, and 8.5 mM, correspondingly. Note the stable node in C and the limit cycle around the unsteady focus in D.

Similar articles

Cited by

References

    1. Jirsa VK, Stacey WC, Quilichini PP, Ivanov AI, Bernard C. On the nature of seizure dynamics. Brain. 2014; 137(8): 2210–2230. - PMC - PubMed
    1. Nagaraj V, Lamperski A, Netoff TI. Seizure Control in a Computational Model Using a Reinforcement Learning Stimulation Paradigm. Int J Neural Syst. 2017; 27(7):1750012 doi: 10.1142/S0129065717500125 - DOI - PubMed
    1. Naze S, Bernard C, Jirsa V. Computational modeling of seizure dynamics using coupled neuronal networks: factors shaping epileptiform activity. PLoS Comput Biol. 2015; 11(5):e1004209 doi: 10.1371/journal.pcbi.1004209 - DOI - PMC - PubMed
    1. Melozzi F, Woodman MM, Jirsa VK, Bernard C. The Virtual Mouse Brain: A Computational Neuroinformatics Platform to Study Whole Mouse Brain Dynamics. eNeuro. 2017; 4(3): pii: ENEURO.0111-17.2017. - PMC - PubMed
    1. Avoli M, D'Antuono M, Louvel J, Kohling R, Biagini G, Pumain R, et al. Network and pharmacological mechanisms leading to epileptiform synchronization in the limbic system in vitro. Prog Neurobiol. 2002; 68(3): 167–207. - PubMed

Publication types

MeSH terms

Substances

Grants and funding

This work was supported by the Russian Science Foundation (project 16-15-10201). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.