1/39 The Fourth International Workshop on Advanced Computation for Engineering Applications (ACEA 2008) MACIS 2 Al-Balqa Applied University, Salt, Jordan Corson Nathalie, Aziz Alaoui M.A. University of Le Havre, LMAH July 24th, 2008
2/39 Biological introduction Table of contents Biological introduction Mathematical models Numerical analysis Conclusion and perspectives
3/39 Biological introduction Neuron a cell body, extended by : dendrites an axon an axon terminal
4/39 Biological introduction Ionic channels Membrane is composed of : a double lipidic layer ionic channels Ionic channels : voltage-dependant open / close / inactive conductance
5/39 Biological introduction Some important channels Sodium channels : 4 gates : 3 to control the opening 1 to control the inactivity particularity : inactivity period Potassium channels : 4 gates all control the opening particularity : delay Leakage channels : particularity : always opened
6/39 Biological introduction Resting potential Potential difference between both sides of the membrane ; more Na + outside more K + inside Experimentally, 90mV < ddp < 50mV intra-cellular space more negative extra-cellular space more positive
7/39 Biological introduction Action potential
8/39 Mathematical models Table of contents Biological introduction Mathematical models Numerical analysis Conclusion and perspectives
9/39 Mathematical models Hodgkin-Huxley model (1) Alan Lloyd Hodgkin and Andrew Fielding Huxley : neurophysiologists experiments on squid giant neuron ; discovery of ionic channels ; description of electric activity of a neuron by mathematical equations ; Nobel price of physiology and medecine in 1952 ;
10/39 Mathematical models Hodgkin-Huxley model (2) C dv = I K + I Na + I L I dt dm = α m (1 m) β m m dt dn = α n (1 n) β n n dt dh = α h (1 h) β h h dt
11/39 Mathematical models Hodgkin-Huxley model (3) I = C dv dt + I K + I Na + I L C dv dt = I K + I Na + I L I I K = n 4 g K (E E K ) ; n = one of the 4 gates of a potassium channel is open ; g K = conductance of a potassium channel number of opened channels ; E K = equilibrium potential of potassium (= Nernst potential) ;
12/39 Mathematical models Hodgkin-Huxley model (4) Similarly, we have : I Na = m 3 hg Na (E E Na ) ; I L = g L (E E L ) ; α i = opening coefficient, i = m, n, h ; β i = closing coefficient, i = m, n, h ;
13/39 Mathematical models Hodgkin-Huxley model (5) C dv = I Na + I K + I L I dt dm = α m (1 m) β m m dt dn = α n (1 n) β n n dt dh = α h (1 h) β h h dt
14/39 Mathematical models Hindmarsh-Rose 1982 model (1) From Hodgkin-Huxley model to Hindmarsh-Rose 1982 model ; Mathematical simplification thanks to biological observations : m is substituted by a constant because of its fast activation ; h + n = 0.8 (experimentally) ; C dv = I K + I Na + I L I dt dm = α m (1 m) β m m dt dn = α n (1 n) β n n dt dh = α h (1 h) β h h dt
15/39 Mathematical models Hindmarsh-Rose 1982 model (2) A Fitzhugh-Nagumo type model : dx dt dy dt = y x 3 + ax 2 + I = 1 dx 2 y a, d : parameters determined experimentally. I : applied current.
16/39 Mathematical models Hindmarsh-Rose 1984 model A third equation to be closer to reality : an adaptation equation dx = y x 3 + ax 2 z + I dt dy = 1 dx 2 y dt dz = ɛ(b(x x c ) z), ɛ << 1 dt slow-fast system
17/39 Numerical analysis Table of contents Biological introduction Mathematical models Numerical analysis Conclusion and perspectives
18/39 Numerical analysis Numerical analysis (1) Time series (t, x) : bursting, and projections of the phase portrait on the planes (x, y), (x, z) et (y, z)
19/39 Numerical analysis Numerical analysis (2) Bifurcation Qualitative change of solutions due to parameters changes I = 1.5: 3-cycle I = 2 : 5-cycle I = 3: 10-cycle 3.25 I 3.295 : chaos I 3.295 : stop generating bursts
20/39 Numerical analysis Numerical analysis (3) I = 1.5, I = 2, I = 2.5, I = 3 I = 3.25, I = 3.33 I = 3.35 I = 3.50
21/39 Table of contents Biological introduction Mathematical models Numerical analysis Conclusion and perspectives
22/39 -Generalities characteristic of many processes in natural systems and non linear science syn : common, chronos : time to share the same motion at the same time there exist many kind of synchronization (identical, generalized, phase...) neurons : optimal transmission of information
Coupling systems (1) Coupling Making different entities be dependant one from the others. Linear coupling function Mutual coupling of 2 neurons by a linear function : electrical connection x 1 = ax1 2 x 1 3 y 1 z 1 k(x 2 x 1 ) y 1 = (a + α)x1 2 y 1 z 1 = µ(bx 1 + c z 1 ) x 2 = ax2 2 x 2 3 y 2 z 2 k(x 1 x 2 ) y 2 = (a + α)x2 2 y 2 z 2 = µ(bx 2 + c z 2 ) 23/39 k(x 1 x 2 ) : coupling function, k : coupling strength
24/39 Coupling systems (2) Sigmoid coupling function Mutual coupling of 2 neurons by a sigmoid function : chemical connection x 1 = ax1 2 x 1 3 y 1 1 z 1 (x 1 V s )k 2 1 + exp( λ(x 2 Θ s )) y 1 = (a + α)x1 2 y 1 z 1 = µ(bx 1 + c z 1 ) x 2 = ax2 2 x 2 3 y 1 2 z 2 (x 2 V s )k 2 1 + exp( λ(x 1 Θ s )) y 2 = (a + α)x2 2 y 2 z 2 = µ(bx 2 + c z 2 ) Θ s : threshold, k 2 : coupling strength.
25/39 Coupling 2 neurons Two neurons identically synchronize if : x 2 x 1 y 2 y 1 z 2 z 1 x = x 1 x 2 0 y = y 1 y 2 0 z = z 1 z 2 0 Stability study of the transverse system : x = x 1 x 2 y = y 1 y 2 z = z 1 Example of theoretical tool to study this system : Lyapunov function. z 2
26/39 HR neurons mutually coupled by a sigmoid function x 1 = ax1 2 x 1 3 y 1 1 z 1 (x 1 V s )k 2 1 + exp( λ(x 2 Θ s )) y 1 = (a + α)x1 2 y 1 z 1 = µ(bx 1 + c z 1 ) x 2 = ax2 2 x 2 3 y 1 2 z 2 (x 2 V s )k 2 1 + exp( λ(x 1 Θ s )) y 2 = (a + α)x2 2 y 2 z 2 = µ(bx 2 + c z 2 )
27/39 Numerical results Remark Diagonal x 1 = x 2 synchronization x 1 vs x 2 for k 2 = 0.8,k 2 = 1,k 2 = 1.22,k 2 = 1.26. Same minimal coupling strength to obtain synchronization between y 1 and y 2 and between z 1 and z 2.
28/39 Coupling 3 neurons x 1 vs x 2 for k 3 = 0.2,k 3 = 0.4,k 3 = 0.6,k 3 = 0.63. same resuts for x i vs x j, y i vs y j, z i vs z j ( i, j = 1, 2, 3)
29/39 Coupling 4 neurons x 1 vs x 2 for k 4 = 0.1,k 4 = 0.3,k 4 = 0.42,k 4 = 0.5. same results for x i vs x j, y i vs y j, z i vs z j ( i, j = 1,..., 4)
30/39 Coupling 5 neurons x 1 vs x 2 for k 5 = 0.1,k 5 = 0.2,k 5 = 0.29,k 5 = 0.3. same results for x i vs x j, y i vs y j, z i vs z j ( i, j = 1,..., 5)
Coupling 6 neurons x 1 vs x 2 for k 6 = 0.1,k 6 = 0.2,k 6 = 0.23,k 6 = 0.24. 31/39 same results for x i vs x j, y i vs y j, z i vs z j ( i, j = 1,..., 6)
Coupling 7 neurons x 1 vs x 2 for k 7 = 0.1,k 7 = 0.17,k 7 = 0.2,k 7 = 0.21. 32/39 same results for x i vs x j, y i vs y j, z i vs z j ( i, j = 1,..., 7)
Coupling 8 neurons x 1 vs x 2 for k 8 = 0.1,k 8 = 0.15,k 8 = 0.16,k 8 = 0.17. 33/39 same results for x i vs x j, y i vs y j, z i vs z j ( i, j = 1,..., 8)
34/39 Coupling n neurons each neuron is connected to all the others in a network of n neurons each neuron is connected to n 1 neurons (n 1 inputs) k n = k 2 n 1 k2 : 2 neurons mutually coupled synchronization threshold n 1 : number of inputs for each neuron
35/39 classical law which is found in many self-organized complex systems
36/39 Examples earthquakes : 1000 of magnitude 4 for 100 of magnitude 5 and 10 of magnitude 6 linguistic : for 1000 occurrences of the in text, 100 occurrences of I and 10 of say urban systems : big cities are rare and small ones are frequent in an exponential way.
37/39 Conclusion and perspectives Table of contents Biological introduction Mathematical models Numerical analysis Conclusion and perspectives
38/39 Conclusion and perspectives Conclusion and perspectives intersection between synchronization in coupled nonlinear systems and complex networks study conditions under which a complex network of dynamical systems synchronizes study of the different topological structures of networks and their consequences on the synchronization phenomenon synchronization and complex networks theoretical study
39/39 Conclusion and perspectives THANK YOU!