Introduction
Epilepsy is a serious neurological disorder characterized by spontaneous recurrent seizures. In the electroencephalogram (EEG) of patients with epilepsy, one may observe seizures (ictal events) and interictal events, e.g., epileptic spikes or spike-waves. These events are characterized by a short duration and rapid onset and offset. The real problem is that about 30 % of the current population of 60 million patients with epilepsy does not respond to any treatment (Kwan and Brodie 2000). This problem is mostly due to an incomplete understanding of the mechanisms that underlie this pathology (e.g., van Drongelen 2007). As is the case for many other neurological diseases, this directly relates to a poor understanding of network function in general. Because of lack of experimental tools for studying network behavior at sufficient scale with the associated detail (e.g., van Drongelen 2010), there is a significant role for modeling in this field (e.g., Blenkinsop et al. 2012;...
References
Abbott LF (2008) Theoretical neuroscience rising. Neuron 60:489–495
Beenhakker MP, Huguenard JR (2009) Neurons that fire together also conspire together: is normal sleep circuitry hijacked to generate epilepsy? Neuron 62:612–632
Benayoun M, Cowan JD, van Drongelen W, Wallace E (2010) Avalanches in a stochastic model of spiking neurons. PLoS Comput Biol 6:e1000846
Berg AT, Berkovic SF, Brodie MJ, Buchhalter J, Cross JH, van Emde Boas W, Engel J, French J, Glauser TA, Mathern GW, Moshé SL, Nordli D, Plouin P, Scheffer IE (2010) Revised terminology and concepts for organization of seizures and epilepsies: report of the ILAE Commission on Classification and Terminology, 2005–2009. Epilepsia 51:676–685
Bi G, Poo M (2001) Synaptic modification by correlated activity: Hebb’s postulate revisited. Annu Rev Neurosci 24:139–166
Blenkinsop A, Valentin A, Richardson MP, Terry JR (2012) The dynamic evolution of focal-onset epilepsies – combining theoretical and clinical observations. Eur J Neurosci 36:2188–2200
Borisyuk RM, Kirilov AB (1992) Bifurcation analysis of a neural network model. Biol Cybern 66:319–325
Chang H-J, Freeman WJ, Burke BC (1998) Optimization of olfactory model in software to give 1/f power spectra reveals numerical instabilities in solutions governed by aperiodic (chaotic) attractors. Neural Netw 11:449–466
Coombes S (2010) Large-scale neural dynamics: simple and complex. Neuroimage 52(3):731–739
Coombes S, Laing C (2009) Delays in activity-based neural networks. Philos Trans R Soc A Math Phys Eng Sci 367(1891):1117–1129
Coombes S, Venkov NA, Shiau L, Bojak I, Liley DTJ, Laing CR (2007) Modeling electrocortical activity through improved local approximations of integral neural field equations. Phys Rev E 76(5):051901
Cosandier-Rimélé D, Merlet I, Bartolomei F, Badier JM, Wendling F (2010) Computational modeling of epileptic activity: from cortical sources to EEG signals. J Clin Neurophysiol 27:465–470
Dayan P, Abbott LF (2001) Theoretical neuroscience computational and mathematical modeling of neural systems. MIT Press, Cambridge, MA
De Schutter E, Bower JM (1994) An active membrane model of the cerebellar Purkinje cell: II. Simulation of synaptic responses. J Neurophysiol 71:401–419
Destexhe A, Sejnowski TJ (2009) The Wilson–Cowan model, 36 years later. Biol Cybern 101:1–2
Douglas RJ, Martin KAC (1991) A functional microcircuit for cat visual cortex. J Physiol 440:735–769
Faye G, Faugeras O (2010) Some theoretical and numerical results for delayed neural field equations. Phys D 239(9):561–578
Foss J, Longtin A, Mensour B, Milton J (1996) Multistability and delayed recurrent loops. Phys Rev Lett 76:708–711
Freeman WJ (1987) Simulation of chaotic EEG patterns with a dynamic model of the olfactory system. Biol Cybern 56:139–150
Goodfellow M, Schindler K, Baier G (2012) Self-organized transients in neural mass models of epileptogenic tissue dynamics. Neuroimage 59:2644–2660
Grimbert F, Faugeras O (2006) Bifurcation analysis of Jansen’s neural mass model. Neural Comput 18:3052–3068
Hebb DO (1949) The organization of behavior. Wiley, New York
Hindriks R, van Putten MJ (2012) Meanfield modeling of propofol-induced changes in spontaneous EEG rhythms. Neuroimage 60:2323–2334
Hopfield JJ (1982) Neurons with graded responses have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA 1:3088–3092
Hutt A, Atay FM (2005) Analysis of nonlocal neural fields for both general and gamma-distributed connectivities. Phys D Nonlinear Phenom 203(1):30–54
Hutt A, Atay FM (2007) Spontaneous and evoked activity in extended neural populations with gamma-distributed spatial interactions and transmission delay. Chaos Solit Fractals 32(2):547–560
Hutt A, Bestehorn M, Wennekers T et al (2003) Pattern formation in intracortical neuronal fields. Netw Comput Neural Syst 14(2):351–368
Izhikevich EM, Edelman GM (2008) Large-scale model of mammalian thalamocortical system. Proc Natl Acad Sci USA 105:3593–3598
Jansen BH, Rit VG (1995) Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. Biol Cybern 73:357–366
Jirsa VK, Haken H (1997) A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics. Phys D 99:503–526
Kwan P, Brodie MJ (2000) Early identification of refractory epilepsy. N Engl J Med 342:314–319
Liley DT, Bojak I (2005) Understanding the transition to seizure by modeling the epileptiform activity of general anesthetic agents. J Clin Neurophysiol 22:300–313
Liley DTJ, Cadusch PJ, Dafilis MP (2002) A spatially continuous mean field theory of electrocortical activity. Netw Comput Neural Syst 13(1):67–113
Little WA (1974) The existence of persistent states in the brain. Math Biosci 19:101–120
Lopes da Silva FH (2008) The impact of EEG/MEG signal processing and modeling in the diagnostic and management of epilepsy. IEEE Rev Biomed Eng 1:143–156
Lopes da Silva FH, Hoeks A, Smits H, Zetterberg LH (1974) Model of brain rhythmic activity. Biol Cybern 15:27–37
Lytton WW (2008) Computer modelling of epilepsy. Nat Rev Neurosci 9:626–637
Lytton WW, Sejnowski TJ (1991) Simulations of cortical pyramidal neurons synchronized by inhibitory neurons. J Neurophysiol 66:1059–1079
Lytton WW, Contreras D, Desthexe A, Steriade M (1997) Dynamic interactions determine partial thalamic quiescence in a computer network model of spike-and-wave seizures. J Neurophysiol 77:1679–1696
Markram H (2006) The blue brain project. Nat Rev Neurosci 7:153–160
May RM (2004) Uses and abuses of mathematics in biology. Science 303:790–793
McCulloch WS, Pitts WH (1943) A logical calculus of the ideas immanent in nervous activity. Bull Math Biophys 5:115–133
Milton J (2000) Epilepsy: multistability in a dynamic disease. In: Walleczek J (ed) Self-organized biological dynamics & nonlinear control. Cambridge University Press, New York, pp 374–386
Milton J (2012) Neuronal avalanches, epileptic quakes and other transient forms of neurodynamics. Eur J Neurosci 36:2156–2163
Milton J, Jung P (2003) Epilepsy as a dynamic disease. Springer, Berlin
Nunez PL (1974) The brain wave equation: a model for the EEG. Math Biosci 21:219–291
Nunez PL (1995) Neocortical dynamics and human EEG rhythms. Oxford University Press, New York
Nunez PL, Srinivasan R (2006a) A theoretical basis for standing and traveling brain waves measured with human EEG with implications for an integrated consciousness. Clin Neurophysiol 117:2424–2435
Nunez PL, Srinivasan R (2006b) Electrical fields of the brain, 2nd edn. Oxford University Press, New York
Olmi S, Livi R, Politi A, Torcini A (2010) Collective oscillations in disordered neural networks. Phys Rev E 81:046119
Rosenblueth A, Wiener N (1945) The role of models in science. In: Philosophy of science, vol 12. The University of Chicago Press, Chicago, pp 316–321
Roxin A, Brunel N, Hansel D (2005) Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks. Phys Rev Lett 94(23):238103
Siekmeier PJ, van Maanen DP (2013) Development of antipsychotic medications with novel mechanisms of action based on computational modeling of hippocampal neuropathology. PLoS ONE 8:e58607
Soltesz I, Staley S (2008) Computational neuroscience in epilepsy. Elsevier, Amsterdam
Strogatz SH (1994) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview Press, Cambridge, MA
Tattini L, Olmi S, Torcini A (2012) Coherent periodic activity in excitatory Erdos-Renyi neural networks: the role of network connectivity. Chaos 22:023133
Touboul J, Wendling F, Chauvel P, Faugeras O (2011) Neural mass activity, bifurcations, and epilepsy. Neural Comput 23:3232–3286
Traub RD, Llinas R (1979) Hippocampal pyramidal cells: significance of dendritic ionic conductances for neuronal function and epileptogenesis. J Neurophysiol 42:476–496
Traub RD, Miles R (1991) Neuronal networks of the hippocampus, vol 777. Cambridge University Press, Cambridge
Traub RD, Jefferys JG, Miles R, Whittington MA, Tóth K (1994) A branching dendritic model of a rodent CA3 pyramidal neurone. J Physiol 481:79–95
Traub RD, Contreras D, Cunningham MO, Murray H, LeBeau FE, Roopun A, Bibbig A, Wilent WB, Higley MJ, Whittington MA (2005) Single-column thalamocortical network model exhibiting gamma oscillations, sleep spindles, and epileptogenic bursts. J Neurophysiol 93:2194–2232
Ullah G, Schiff SJ (2009) Models of epilepsy. Scholarpedia 4(7):1409
van Albada SJ, Robinson PA (2009) Mean-field modeling of the basal ganglia-thalamocortical system. I Firing rates in healthy and parkinsonian states. J Theor Biol 257:642–663
van Drongelen W (2007) Guest editorial. J Clin Neurophysiol 24:85–86
van Drongelen W (2010a) Signal processing for neuroscientists, a companion volume: advanced topics, nonlinear techniques and multichannel analysis. Elsevier, Amsterdam
van Drongelen W (2010b) Tools for epilepsy research. J Clin Neurophysiol 27:359
van Drongelen W (2013) Modeling neural activity. ISRN Biomath 2013:ID 871472
van Drongelen W, Lee HC, Hereld M, Jones D, Cohoon M, Elsen F, Papka ME, Stevens RL (2004) Simulation of neocortical epileptiform activity using parallel computing. Neurocomputing 58–60:1203–1209
van Drongelen W, Lee HC, Koch H, Hereld M, Elsen F, Chen Z, Stevens RL (2005) Emergent epileptiform activity in neural networks with weak excitatory synapses. IEEE Trans Neural Syst Rehabil 13:236–241
van Drongelen W, Koch H, Elsen FP, Lee HC, Mrejeru A, Doren E, Marcuccilli CJ, Hereld M, Stevens RL, Ramirez JM (2006) The role of persistent sodium current in bursting activity of mouse neocortical networks in vitro. J Neurophysiol 96:2564–2577
van Drongelen W, Lee HC, Stevens RL, Hereld M (2007) Propagation of seizure-like activity in a model of neocortex. J Clin Neurophysiol 24:182–188
van Drongelen W, Martell A, Lee HC (2008) Neocortical epileptiform activity in neuronal models with biophysically realistic ion channels. In: Soltesz I, Staley S (eds) Computational neuroscience in epilepsy. Elsevier, Amsterdam, pp 168–183
van Gils SA, Janssens SG, Kuznetsov YA, Visser S (2013) On local bifurcations in neural field models with transmission delays. J Math Biol 66:837–87
van Rotterdam A, Lopes da Silva FH, van den Ende J, Viergever MA, Hermans AJ (1982) A model of the spatial-temporal characteristics of the alpha rhythm. Bull Math Biol 44:283–305
van Vreeswijk CA (1996) Partial synchronization of populations of pulse-coupled oscillators. Phys Rev E54:5522
van Vreeswijk CA, Abbott LF, Ermentrout GB (1994) Inhibition, not excitation, synchronizes coupled neurons. J Comput Neurosci 1:303–313
Veltz R (2011) An analytical method for computing Hopf bifurcation curves in neural field networks with space-dependent delays. C R Math Acad Sci Paris 349(13–14):749–752
Veltz R, Faugeras O (2010) Local/global analysis of the stationary solutions of some neural field equations. SIAM J Appl Dyn Syst 9(3):954–998
Venkov NA, Coombes S, Matthews PC (2007) Dynamic instabilities in scalar neural field equations with space-dependent delays. Phys D Nonlinear Phenom 232(1):1–15
Visser S, Meijer HGE, Lee HC, van Drongelen W, van Putten MJAM, van Gils SA (2010) Comparing epileptiform behavior of meso-scale detailed models and population models of neocortex. J Clin Neurophysiol 27:471–478
Visser S, Meijer HGE, van Putten MJAM, van Gils SA (2012) Analysis of stability and bifurcations of fixed points and periodic solutions of a lumped model of neocortex with two delays. J Math Neurosci 2:8
Vogels TP, Sprekeler H, Zenke F, Clopath C, Gerstner W (2011) Inhibitory plasticity balances excitation and inhibition in sensory pathways and memory networks. Science 334:1569–1573
Volman V, Bazhenov M, Sejnowski TJ (2012) Computational models of neuron-astrocyte interaction in epilepsy. Front Comput Neurosci 6:1–10
Wallace E, Benayoun M, van Drongelen W, Cowan JD (2011) Emergent oscillations in networks of stochastic spiking neurons. PLoS ONE 6(5):e14804
Wendling F, Chauvel P (2008) Transition to ictal activity in temporal lobe epilepsy: insights from macroscopic models. In: Soltesz I, Staley K (eds) Computational neuroscience in epilepsy. Elsevier, Amsterdam, pp 356–387
Wendling F, Bartolomei F, Mina F, Huneau C, Benquet P (2012) Interictal spikes, fast ripples and seizures in partial epilepsies – combining multi-level computational models with experimental data. Eur J Neurosci 36:2164–2177
Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24
Wilson HR, Cowan JD (1973) A mathematical theory of the functional dynamics of nervous tissue. Biol Cybern 13:55–80
Woodin MA, Ganguly K, Poo MM (2003) Coincident pre- and postsynaptic activity modifies GABAergic synapses by postsynaptic changes in Cl− transporter activity. Neuron 39:807–820
Further Reading
Coombes S (2005) Waves, bumps, and patterns in neural field theories. Biol Cybern 93(2):91–108
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this entry
Cite this entry
van Gils, S., van Drongelen, W. (2013). Epilepsy: Computational Models. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7320-6_504-1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7320-6_504-1
Received:
Accepted:
Published:
Publisher Name: Springer, New York, NY
Online ISBN: 978-1-4614-7320-6
eBook Packages: Springer Reference Biomedicine and Life SciencesReference Module Biomedical and Life Sciences