School on Modelling, Automation and Control of Physiological variables at the Faculty of Science, University of Porto 2-3 May, 2007 Topics on Biomedical Systems Modelling: transition to epileptic activity Fernando Lopes da Silva The case of the brain rhythm in the alpha (8-12 Hz) frequency range appearing over the somatosensory cortex: the mu rhythm Center of Neuroscience Swammerdam Institute for Life Sciences University of Amsterdam One interesting property of the Alpha rhythm of the somatosensory cortex (Mu rhythm) is that it is modulated by movements of the hands: It is attenuated, desynchronized alpha (ERD), by moving the hands, and it is enhanced at rest (synchronized alpha or ERS) Power changes in the Apha (Mu rhythm) and Beta frequency ranges ERD and ERS Depending on whether the hand or the foot is moved, the spatio-temporal pattern differs; the patterns of both conditions appear as mirror images. Two questions put to the theoreticians or modelers: 1. How is the Alpha rhythmic activity generated? 2. How can the mirror images of the spatio-temporal patterns of ERD/ERS associated with hand or foot movements be accounted for?
Basic neuronal network responsible for rhythmic activities in thalamo-cortical circuits Computational model of the thalamocortical neuronal networks Steriade 1999 Basic equations of the model (1) Time evolution of the neuronal membrane potential: Synaptic currents Synaptic conductances are modeled by convolving firing rate frequency with synaptic impulse response Nonlinear GABA-B synaptic response Nonlinearity is realized by a sigmoidal function of the form: The model was realized using the Simulink toolbox of Math Works. Simulations were run using the ode3 integration method with a time step of 1 millisecond duration. Postprocessing was done using Matlab. Transfer between firing rate and membrane potential Basic equations of the model (2) Transfer function for the burst firing mode Where G B is the maximal firing rate within a burst, variables n inf (V) and m inf (V) are static sigmoidal functions that describe the fractions of neurons that are deinactivated or activated, respectively. Expressions (9) and (10) describe the time delay of I T inactivation.
Model scheme This result is an answer to question # 1: How is the Alpha rhythmic activity generated? pyramidal cells population interneuronal population thalamocortical cells population thalamic RE cells population external inputs burst generation process SEIN, 2003 Thalamocortical network pyramidal cell GABAergic interneuron But we have to consider also the second question: 2. How can the mirror images of the spatiotemporal patterns of ERD/ERS associated with hand or foot movements be accounted for? Thalamic Reticular Nucleus Thalamocortical Relay Nucleus thalamic reticular (RE) neuron thalamocortical (TC) neuron Extracellular activity of a RE neuron (yellow) and cortical field potential (green) recorded in the GAERS during a spike and wave discharge downloaded from Crunelli Research Group: www. thalamus.org.uk Medical Physics Department Excitation Inhibition In both TC and RE cells burst firing is provided SEIN, 2003 by I T calcium current ERS ERD This result means that the mechanism of recurrent inhibition between neighboring thalamo-cortical modules can account for the mirror images of the spatio-temporal patterns of ERD/ERS elicited by hand or foot movements, respectively.
How can this transition to epileptic activity take place? How does the transition to epileptic activity take place? We have to examine how this occurs in patients and in animal models The WAG/Rij rat as model for absences seizures (Gilles van Luijtelaar and Ton Coenen) EEG and Video during an epileptic absence ( petit mal )! genetic model.! no neurological defects.! absences are characterized by behavioral arrest and spike and wave discharges (SWDs) in the EEG.! pharmacological responses is similar to that of patients with absences. Spontaneous absence: Patient is requested to press a button immediately after a technician did the same. These observations indicate that neuronal networks can display qualitatively different dynamical states. This is likely to be due to the fact that these neuronal networks are complex non-linear systems: Such networks may display complex dynamics with more than one stable state; in this case: " A normal on-going steady-state, and " An oscillatory epileptiform, or paroxysmal state. This is what happens in epilepsy.
Ca 2+ T-channel GABA A & B Computer model of a thalamocortical network capable of displaying a bifurcation betweem two states, (i) a normal oscillatory state, and (ii) a paroxystic seizure state. On-going state- model Spindle - rat Simulation example Simulated epoch Power spectra Paroxysmal state - model Paroxysm - rat This is evidence for bi-stability: one network two stable states Example of a bifurcation between two states: normal & seizure (absence type), both in the model and in EEG real signals. Sensitivity of the Model to a set of parameters Occurrence of transition to epileptic seizure mode: parameter sensitivity Phase portraits of the system under non epileptic and epileptic conditions
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 What are the predictions of the model with respect to the dynamics of absence seizures? " One prediction is that for this kind of seizures the transition occurs randomly; " A second prediction is that it should be possible to stimulate the system in such a way that the transition to the seizure mode may be aborted. This implies that it should be possible to control the system s behavior. The 1st prediction was tested by calculating the distributions of durations and of intervals inter-paroxysms. Distribution of Durations either of paroxysmal events or of interparoxysmal events Distributions of epochs duration - comparison of simulated and rat (WAG/Rij) experimental data In common language: Termination of a process is random in time with constant probability simple calculation Prediction Exponential distribution of process durations In math language: Probability of termination in unit time : p Probability of survival of unit time : 1- p P(t) = (1-p)(1-p).(1-p)p 1 - p = e -!! p = 1 - e -! P(t) = (1 - e -! )e -!t e -! " 1 -! P(t) =!e -!t Number of processes!e -!t log time Process duration Medical Physics Department SEIN, 2003 Medical Physics Department SEIN, 2003 Quasi- exponential (a ~ 1) distribution of SWDs in rat (WAG/Rij) Quasi-exponential distribution of duration of 3 Hz paroxysms in a patient with absence nonconvulsive seizures during the night
Gamma distribution of SWDs duration of GAER rats But. Does it hold in all similar cases? Not exactly. " # 1 # x /! y = Cx e Thus, what do we have to modify in the model? It is necessary to include a use-dependent parameter, i.e. a parameter that changes as a seizure progresses. New hypothesis to be tested: K+ accumulation occurs in the course of SWD in glial cells affecting the excitability of neurons., y = Cx e " # 1 # x /! Real EEG signals Neuronal networks Statistics/ Dynamics Signal analysis The second prediction is that it should be possible to control the occurrence or the evolution of a seizure by means of counterstimulation. Models/Simulated EEGs Statistics/ Dynamics Indeed in bistable systems a limit cycle may be annihilated by a perturbation applied at the appropriate time.
Counter-stimulation is capable of annihilating the transition to the paroxysmal oscillation Negative stimulus Positive stimulus Collaborators from the Institute of Epilepsy SEIN ( Meer en Bosch, Heemstede) and MEG Center (Free University, Amsterdam): Stiliyan Kalitzin, Piotr Suffczynski Jaime Parra. Dimitri Velis. Wouter Blanes. Elan Ohayon Fernando Lopes da Silva Suffczynski P, Lopes da Silva FH, Parra J, Velis DN, Bouwman BM, van Rijn CM, van Hese P, Boon P, Khosravani H, Derchansky M, Carlen P, Kalitzin S. Dynamics of epileptic phenomena determined from statistics of ictal transitions. IEEE Trans Biomed Eng. 2006 Mar;53(3):524-32. Suffczynski P, Lopes da Silva F, Parra J, Velis D, Kalitzin S. Epileptic transitions: model predictions and experimental validation. J Clin Neurophysiol. 2005 Oct;22(5):288-99. Suffczynski P, Kalitzin S, Lopes da Silva FH. Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network. Neuroscience. 2004;126(2):467-84. Lopes da Silva F, Blanes W, Kalitzin SN, Parra J, Suffczynski P, Velis DN. Epilepsies as dynamical diseases of brain systems: basic models of the transition between normal and epileptic activity. Epilepsia. 2003;44 Suppl 12:72-83. Lopes da Silva FH, Blanes W, Kalitzin SN, Parra J, Suffczynski P, Velis DN. Dynamical diseases of brain systems: different routes to epileptic seizures. IEEE Trans Biomed Eng. 2003 May;50(5):540-8. Suffczynski P, Kalitzin S, Pfurtscheller G, Lopes da Silva FH. Computational model of thalamo-cortical networks: dynamical control of alpha rhythms in relation to focal attention. Int J Psychophysiol. 2001 Dec;43(1):25-40. Thalamo-cortical networks possess bi-stability. In Phase-space: the normal steady-state is within the separatrix ( ), the complex oscillatory (paroxysmal) state is outside.