Multimodal Oscillations: from Dopamine Neurons to Solid Fuel Combustion. Georgi Medvedev. Department of Mathematics, Drexel University

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1 Mulimodal Oscillaions: from Dopamine Neurons o Solid Fuel Combusion Georgi Medvedev Deparmen of Mahemaics, Drexel Universiy

2 Dopamine Neurons and Solid Fuel Wha? A mahemaician is he one, who finds analogies beween saemens; a beer mahemaician is he one, who finds analogies beween proofs; ye a beer mahemaician is he one, who finds analogies beween heories; bu one can also imagine such (mahemaician), who finds analogies beween analogies. Sephan Banach An applied mahemaician is he one, who finds analogies beween models.

3 Mulimodal Oscillaions: Examples a b c v, v v, v v, v Ellipic bursers (Rinzel, 987; Rinzel, Ermenrou, 989; Rubin, Terman, 22) Mixed-mode bursing, I NaP + I KS model (Wang, 993) A model of he dopamine neuron (Medvedev, Cisernas, Physica D, 24; Medvedev, Wilson, Callaway, and Kopell, JCNS, 23) A model of he sellae cell (Rosein, Oppermann, Whie, Kopell, JCNS, 26) Chemical oscillaions: Belousov-Zhaboinskii reacion, combusion

4 Mulimodal Oscillaions in a Model of he Dopamine Neuron a b c v, v v, v v, v a SN PD a Period v τ PD AHB b Firing number τ Medvedev, Cisernas, Physica D, 24

5 Inerspike inerval hisograms (noise inensiy: σ =.) Inerspike inerval disribuions 45 τ=9.4 σ=. 35 τ=9.8 σ=. 8 τ=.2 σ=. 8 τ=.5 σ= Volage ime series σ=. τ=9.4 σ=. τ=9.8 σ=. τ=.2 σ=. τ= v v v v

6 Experimenal Daa ( couresy of C.J. Wilson, UT, San Anonio)

7 Single Comparmen Model Curren balance equaion: C dv d = g Ca(v)(E Ca v) + g KCa (u)(e K v) + g l (E l v) Nonlinear conducances g Ca (v) and g KCa (u): 8 x g Ca, ms/cm g KCa,mS/cm a v, mv b u, nm Ca-equaion: du d du d du d = [Ca]-influx [Ca]-efflux volume volume = I Ca surface area = ω volume { g Ca (v) (E Ca v) u τ [Ca] surface area τ Ca volume }

8 Phase Plane for a Single Comparmen Model dv d = C (g Ca(v)(E Ca v) + g KCa (u)(e K v) + g l (E l v)) du d = ω ( g Ca (v) (E Ca v) u ) τ 3 v 4 u v, mv , s b u, nm c, s ω - naural frequency of oscillaions Medvedev, Kopell, SIAP, 2

9 Mulicomparmen model Geomery: ω ω ω 2 3 { dvi = d C (g Ca(v i )(E Ca v i ) + g KCa (u i )(E K v i ) + g l (E l v i ) + d(v i+ 2v i + v i ) du i ( ) = ω d i gca (v i ) (E Ca v i ) u i τ, i =, 2,..., N u, nm u, nm c v, mv d v, mv 5

10 Individual oscillaors are near Andronov-Hopf Bifurcaion

11 Individual oscillaors are near Andronov-Hopf Bifurcaion K K CLOSE TO HOPF BIFURCATION a v, v Medvedev, Cisernas, Physica D, 24

12 Reducion o a D mapping Winding number λ P (ξ) = ( αω T (ξ)) ξ + αω 2 T (ξ) f(ξ), α = τ

13 Mulimodal Oscillaions in a Model of he Dopamine Neuron a b c v, v v, v v, v n a Period 3 2 v [V] branch of fixed poins τ Ca [µm/sec] b Firing number τ Medvedev, Cisernas, Physica D, 24

14 A Free Boundary Problem of Condensed Phase Combusion FUEL x=s() PRODUCT u (x, ) = u xx (x, ), u (s(), ) = g (ṡ()), < x < s() u x (s(), ) = ṡ() Kineic funcion: g(ξ) = + νk( ξ), k p ( ξ) = ξp ξ p+ Conrol parameers: ν, p Traveling wave soluion: u(x, ) = e x+, x u x= x

15 The Frankel-Royburd Model of Solid Fuel Combusion A 3D approximaion of a free boundary problem modeling solid combusion (Frankel, Kova cic, Royburd, Timofeev, Physica D, 2) v = 3(v 3+v 2 v ) νk(v ) v 2 νk (v ) v 2 = v 3 v v 3 = 9(v v 3 ) 6v 2 + ν(v + )k(v ) + 2v 2 Kineic funcion: k(v) = ( v)p ( v), p + Conrol parameers: ν, p

16 Oscillaory paerns Supercriical Andronov-Hopf bifurcaion Subcriical Andronov-Hopf bifurcaion. p=2.2 α=..4 p=2.2 α=..5 p=3 α=.5.5 p=3 α= p=2.2 α=..3 p=2.2 α=. p=3 α=.5 p=3 α=.5 x x x x Medvedev, Yoo, Physica D, 27.

17 Mixed-mode oscillaions.6 p=3 α= Σ Π Σ x 2 ISI α ẋ = α b(α) b(α) α x + h(x, α)

18 Three specific quesions Frequency doubling.5 p=2.2 α=.. p=2.2 α=. x Long inerspike inervals.5 p=3 α=.5.5 p=3.4 γ= α=.45.5 p=2.77 γ=.34 α= Chaos a he border of beween sub- and supercriical Andronov-Hopf Bifurcaion

19 The Transiion from Sub- o Supercriical Andronov-Hopf Bifurcaion.5 p=3 α=.5.5 p=2.2 α=. x

20 The Transiion from Sub- o Supercriical Andronov-Hopf Bifurcaion p=3.4 γ= α=.45 p=2.77 γ=.34 α=.45 p=2.76 γ=.766 α=.45 p=2.736 γ=.33 α= p=2.72 γ=.28 α=.45 p=2.69 γ=.44 α=.45 p=2.67 γ=.542 α=.45 p=2.6 γ=.327 α= x 3 x 3 x 3 x x x x x.5

21 Analysis.2.6 x Σ Π Σ x x Rescale and swich o cylindrical coordinaes: (x,, x 3 ) (ɛ 2 ξ, ɛρ cos θ, ɛρ sin θ), α = µɛ 2 Slow manifold: S = { (x, ρ, θ) : x = U(θ)ρ 2, ρ 2ɛM }, U(θ) = a + A cos (2θ ) Reducion: { İ = 2ɛ 2 (µi + γ + Q(φ)) + O(ɛ 3 ), φ = + O(ɛ), I = r 2

22 Frequency doubling: he geomery of he periodic orbi.5 p=2.2 α=.. p=2.2 α=. x x(θ) = ( ρ 2 (a + A cos 2θ), ρ cos θ, ρ sin θ ), ρ = α, θ [, 2π) γ.5 µ=.4 µ=.4.2 µ=.4 x 3 x x x k(θ) = [ẋ, ẍ] ẋ 3 = + 2A 2 (3 cos 4θ + 5) κ(θ) = (ẋ, ẍ, ẍ) [ẋ, ẍ] 2 = γ α 6A sin 2θ + 2A 2 (3 cos 4θ + 5) θ [, 2π).

23 The Inerspike Inervals Long inerspike inervals.5 p=3 α=.5.5 p=3.4 γ= α=.45.5 p=2.77 γ=.34 α= τ τ i τ +, τ = ( 2α ln + α ) γ(p) C ɛ 4,

24 Chaos a he border of criicaliy: reducion o a D mapping.2 p=3 α=.5 x 3.2 I n+ γ> γ< x I n 2 I(φ) = ρ 2, { İ = 2ɛ 2 (µi + γ + Q(φ)) + O(ɛ 3 ), φ = + O(ɛ), I n+ = G (I n ), G (I) = ( 2αω) I 2ɛ 2 ωγ Fixed poin: Ī = γ α

25 Chaos a he border of criicaliy 8 8 I n+ 4 I n I n 4 8 I n.4 p=2.72 γ=.28 α=.45 p=2.76 γ=.766 α=

26 Chaoic mixed-mode oscillaions in he Hodgkin-Huxley model v (mv) (ms) C v = g Na m 3 (v)h (v E Na ) g K n 4 (v E K ) g l (v E l ) τ n ṅ = n (v) n τ h ḣ = h (v) h Conrol parameers: τ n, τ h Medvedev, Yoo, arxiv:

27 Reducion o a D map spike.6 η 2.2 x n+.2 x n η ξ.6.2 x n.6.2 x n I ξ η η 2 = λ α β β α ξ η η 2 + N (ξ, η), The Poincare map: P : x( i ) x( i+ ) x = α(η 2 + η2 2 ) P (x) = x 2αω(x + γ) + O(α 2 ), ω = 2πβ,

28 Transiion o mixed-mode oscillaions x n v (mv) EN τn 2 Frequency, % x n I (ms) τ n 3 5 Number of small oscillaions Number of spikes in one burs is disribued geomerically EN τn a ( τ n τ n )

29 Transiion from onic spiking o bursing v γ=2. v γ= v γ=2.93 v γ= Medvedev, PRL 26

30 Transiion o bursing x L Σ x L Σ E E y sn y c y bp y y sn y c y bp y

31 Transiion o bursing.6.5 P(w) v γ= w I w J A criical value γ = γ c : max w I P γ c(w) = w HC Assume ha map P γ c : I I has a. an invarian probabiliy measure µ, b. he mixing propery: lim n µ ( A P n γ c (B)) = µ(a)µ(b), A, B I.

32 Saisical properies of he irregular bursing γ= γ= P(w).4 N γ T γ 3 I J w γ γ Denoe δ = µ(j), for small γ γ c >. Then δ γ = { O ( ) (γ γ c ) 2, γ c < γ < γ, O (γ γ c ), γ > γ; N γ = O ( δ ) = { O ) ((γ γ c ) 2, γ c < γ < γ, O ( (γ γ c ) ), γ > γ. T γ = { T γ c O ( ) γ γ c, T O ((γ γ c ) ln (γ γ c ) ), γ c < γ < γ, γ > γ. Medvedev, PRL, 26

33 Conclusions We analyzed mechanisms for mixed-mode oscillaions in he model of a dopamine neuron, he finie dimensional model of solid fuel combusion. References: G.S. Medvedev and N. Kopell, Synchronizaion and ransien dynamics in he chains of elecrically coupled FizHugh-Nagumo oscillaors, SIAM J. Appl. Mah., vol. 6, No. 5, pp G.S. Medvedev and J. Cisernas, Mulimodal regimes in a comparmenal model of he dopamine neuron, Physica D, 24 G.S. Medvedev and Y. Yoo, Mulimodal oscillaions in sysems wih srong conracion, Physica D, 228(2), 87-6, 27. G.S. Medvedev and Y. Yoo, Chaos a he border of criicaliy, arxiv: G.S. Medvedev, Transiion o bursing via deerminisic chaos, Phys. Rev. Le., 26 Acknowledgmens: This work was parially suppored by he Naional Science Foundaion under Gran No