Nerve cell model and asymptotic expansion Yasushi ISHIKAWA - PDF document

Nerve cell model and asymptotic expansion Yasushi ISHIKAWA [Department of Mathematics, Ehime University (Matsuyama)] 1 1 Analysis on the Wiener-Poisson space 1.1 SDE on the Wiener-Poisson space evy process, R m -valued, with L´ Let z ( t ) be a L´ evy measure µ ( dz ) such that the character- istic function ψ t is given by 1 � ( e i ( ξ,z ) − 1 − i ( ξ, z ) ψ t ( ξ ) = E [ e i ( ξ,z ( t )) ] = exp( t 1 + | z | 2 ) µ ( dz )) . We may write � t 1 � z ( t ) = ( z 1 ( t ) , ..., z m ( t )) = z { N ( dsdz ) − 1 + | z | 2 .µ ( dz ) ds } , m \{ 0 } R 0 where N ( dsdz ) is a Poisson random measure on T × ( R m \ { 0 } ) with mean ds × µ ( dz ). We define a jump-diffusion process X t by an SDE � t � t � t � g ( X s − , z ) ˜ X t = x + b ( X s − ) ds + σ ( X s − ) dW ( s ) + N ( dsdz ) . ( ∗ ) m \{ 0 } R 0 0 0 Here, b ( x ) = ( b i ( x )) is a continuous functions on R d , Lipschits continuous and R d valued, σ ( x ) = ( σ ij ( x )) is a continuous d × m matrix on R d , Lipschits continuous, and g ( x, z ) is a continuous functions on R d × R m and R d valued, We assume | b ( x ) | ≤ K (1 + | x | ) , | σ ( x ) | ≤ K (1 + | x | ) , | g ( x, z ) | ≤ K ( z )(1 + | x | ) , and | b ( x ) − b ( y ) | ≤ L | x − y | , | σ ( x ) − σ ( y ) | ≤ L | x − y | , | g ( x, z ) − g ( y, z ) | ≤ L ( z ) | x − y | . Here K, L are positive constants, and K ( z ) , L ( z ) are positive functions satisfying � { K p ( z ) + L p ( z ) } µ ( dz ) < + ∞ , R m \{ 0 } where p ≥ 2. We shall introduce assumptions concerning the L´ evy measure. Set � | z | 2 µ ( dz ) . ϕ ( ρ ) = | z |≤ ρ 1 Some parts of this talk are based on joint works with Dr. M. Hayashi and with Prof. H. Kunita. 1

We say that the measure µ satisfies an order condition if there exists 0 < α < 2 such that ϕ ( ρ ) lim inf > 0 . ρ α ρ → 0 We assume µ satisfies the order condition. We make a perturbation of the trajectory F = ξ s , 0 < s < t by ε + u , which will be introduced in Sect. 1.2. Let u = ( u 1 , ..., u k ). Then ξ s,t ( x ) ◦ ε + u = ξ t k ,t ◦ φ z k ◦ ξ t k − 1 ,t k ◦ φ z k − 1 · · · ◦ φ z 1 ◦ ξ s,t 1 ( ∗∗ ) Example of SDE (Linear SDE) X t is given by m � dX t = V 0 ( X t − ) dt + V j ( X t − ) dz j ( t ) . (1 . 1) j =1 Here V 0 , V 1 , ..., V m are smooth vector fields on R d . In what follows we assume b ( x ) and σ ( x ) are g ( x, z ) functions having bounded deriva- tives of all orders, for simplicity. 1.2 The nondegeneracy condition We say that F satisfies the (ND) condition if for all p ≥ 1 , k ≥ 0 there exists β ∈ ( α 2 , 1] such that p � � � − 1 � � � � | ( v, ˜ D u F |≤ ρ β } ˆ ( v, Σ v ) + ϕ ( ρ ) − 1 D u F ) | 2 1 {| ˜ ◦ ǫ + � � sup sup sup E [ N ( du ) ] < ∞ , � τ � � A (1) � ρ ∈ (0 , 1) d τ ∈ A k ( ρ ) v ∈ R , � � | v | =1 � where Σ is the Malliavin’s covariance matrix Σ = (Σ i,j ), where Σ i,j = ( D t F i , D t F j ) dt . T More precisely we can prove the following result. Proposition 1 Let F ∈ D ∞ satisfy the (ND) condition. For any m there exist k, l , p > 2 and C m > 0 such that | ϕ ◦ F | ′ � | (1 + | F | 2 ) β | k,l,p ) � ϕ � − 2 m k,l,p ≤ C m ( (1 . 9) β ≤ m for ϕ ∈ S . Here | F | k,l,p is a 3-parameters Sobolev norm on the Wiener-Poisson space, and | F | ′ k,l,p is its dual norm given above. � ϕ � − 2 m is a norm introduced on S as follows. 2

Let S be the set of all rapidly decreasing C ∞ -functions and let S ′ be the set of tempered distributions. For ϕ ∈ S we introduce a norm � 1 � {| (1 − ∆) β (1 + | y | 2 ) α ϕ | 2 } dy ) � ϕ � 2 m = ( (1 . 10) 2 | α | + | β |≤ m for m = 1 , 2 , ... . We let S 2 m to be the completion of S with respect to this norm. We remark S ⊂ S 2 m , m = 1 , 2 , .. . We introduce the dual norm � ψ � − 2 m of � ϕ � 2 m by � ψ � − 2 m = sup | ( ϕ, ψ ) | , (1 . 11) ϕ ∈S 2 m , � ϕ � 2 m =1 � ϕ ( x ) ¯ where ( ϕ, ψ ) = ψ ( x ) dx . We denote by S − 2 m the completion of S with respect to the norm � ψ � − 2 m . Further, put S ∞ = ∩ m ≥ 1 S 2 m , S −∞ = ∪ m ≥ 1 S − 2 m By (1.9) we can extend ϕ to T ∈ S − 2 m . Since ∪ m ≥ 1 S − 2 m = S ′ , we can define the composition T ◦ F for T ∈ S ′ as an element in D ′ ∞ . A sufficient condition for the composition using the Fourier method is the (ND) condition stated above. 1.3 The (UND) condition We state a sufficient condition for the asymptotic expansion so that Φ ◦ F ( ǫ ) can be expanded as ∞ 1 � � Φ ◦ F ( ǫ ) ∼ n !( ∂ n Φ) ◦ f 0 · ( F ( ǫ ) − f 0 ) n m =0 | n | = m ∼ Φ 0 + ǫ Φ 1 + ǫ 2 Φ 2 + ... in D ′ ∞ for Φ ∈ S ′ , under the assumption that ∞ ǫ j f j ∼ f 0 + ǫf 1 + ǫ 2 f 2 + ... in D ∞ . � F ( ǫ ) ∼ m =0 Definition 1 We say F ( ǫ ) = ( F 1 , . . . , F d ) satisfies the (UND) condition (uniformly non-degenerate) if for any p ≥ 1 and any integer k it holds that lim sup sup sup ess sup E [ | (( v, Σ( ǫ ) v ) ǫ → 0 ρ ∈ (0 , 1) v ∈ R d τ ∈ A k ( ρ ) | v | =1 � N ( du )) − 1 ◦ ǫ + + ϕ ( ρ ) − 1 | ( v, ˜ D u F ( ǫ )) | 2 1 {| ˜ D u F ( ǫ ) |≤ ρ β } ˆ τ | p ] < + ∞ , A (1) T D t F i ( ǫ ) D t F j ( ǫ ) dt . where Σ( ǫ ) = (Σ i,j ( ǫ )) , Σ i,j ( ǫ ) = � Note that these are not formal expansions, but they are asymptotic expansions with respect to the norms | . | k,l,p and | . | ′ k,l,p respectively. 3

Proposition 2 (Hayashi-I [3]) Suppose F ( ǫ ) satisfies the (UND) condition, and that F ( ǫ ) ∼ � ∞ j =0 ǫ j f j in D ∞ . Then, for all Φ ∈ S ′ , we have Φ ◦ F ( ǫ ) ∈ D ′ ∞ has an asymptotic expansion in D ′ ∞ : ∞ 1 � � Φ ◦ F ( ǫ ) ∼ n !( ∂ n Φ) ◦ f 0 · ( F ( ǫ ) − f 0 ) n m =0 | n | = m ∼ Φ 0 + ǫ Φ 1 + ǫ 2 Φ 2 + ... in D ′ ∞ . Here Φ 0 , Φ 1 , Φ 2 are given by the formal expansion d � f i Φ 0 = Φ ◦ f 0 , Φ 1 = 1 ( ∂ x i Φ) ◦ f 0 , i =1 d d 2 ( ∂ x i Φ) ◦ f 0 + 1 1 f j � f i � f i 1 ( ∂ 2 Φ 2 = x i x j Φ) ◦ f 0 , 2 i =1 i,j =1 d d d 3 ( ∂ x i Φ) ◦ f 0 + 2 x i x j Φ) ◦ f 0 + 1 1 f j 2 ( ∂ 2 1 f j 1 ( ∂ 3 � f i � f i � f i 1 f k Φ 3 = x i x j x k Φ) ◦ f 0 2! 3! i =1 i,j =1 i,j,k =1 ... 4

2 Application to the asymptotic expansion 2.1 Neuron cell model : H-H ( Hodgkin-Huxley ) model A H-H model is a biological model of an active nerve cell (neuron) which produces spikes (electric bursts) according to the input from other neurons. The simplified H-H model, called Fitzhugh-Nagumo model, is described in terms of ( V ( t ) , n ( t )) by ∂V ∂t ( t, V ( t )) = − g · h ( V ( t ) − E N a ) − n 4 ( t, V ( t ) − E K ) + I ( t ) , ∂n ∂t ( t, V ( t )) = α ( V ( t ))(1 − n ( t )) − β ( V ( t )) n ( t ) . ( ∗ ) Here h ( . ) denotes “conductance” of the natrium (sodium) ion channel, g is a coefficient, I ( . ) is an input current, and n ( t ) = n ( t, v ( t )) denotes the depolarization rate (permeability) of potasium ion chanel. Constants E Na , E K correspond to standstill electric potential due to natrium, potasium ions respectively, and α ( . ) and β ( . ) are some functions describing the transition rate from closed potasium channel to open potasium channel (open potasium channel to closed potasium channel), respectively. In case we study the chain of nerve cells, we have to take into consideration the effect of transmission of external signals and noise throgh synapse. To this end we introduce a jump-diffusion process z ( t ) to model the signal and noise. A stochastic model V ( t, ǫ ) in this case is described by the SDE dV ( t, ǫ ) = − g · h ( V ( t ) − E N a ) dt − n 4 ( t, V ( t ) − E K ) dt + � A i 1 { t i } ( t ) + I ( t ) dt + ǫdz ( t ) , t i ≤ t dn ( t ) = α ( V ( t ))(1 − n ( t )) dt − β ( V ( t )) n ( t ) dt. ( ∗∗ ) Here ( t i ) denotes the arrival times of external spikes, ǫ > 0 is a parameter, and dz ( t ) denotes a stochastic integral with respect to the noise process z ( t ). In this model we take z ( t ) to be a jump-diffusion; the diffusion part corresponds to the continuous noise (i.e. white noise), and the jump part corresponds to the discontinuous noise. The reason for taking such kind of noise is that the transmission of information among nerve cells are due to chemical particles (synaptic vesicles) which may induce discontinuous random effects. We construct the following model; the pair ( V ( t ) , n ( t )) is denoted by ( X ǫ t , Y ǫ t ). � t � t X ǫ c 1 ( X ǫ s , Y ǫ I ( s ) ds + Z ǫ � t = x 0 + s ) ds + A i 1 { t i } ( t ) + t , 0 0 t i ≤ t � t Y ǫ c 2 ( X ǫ s , Y ǫ t = y 0 + s ) ds. (2 . 1) 0 Here � x 2 + y 2 ) x − y, c 1 ( x, y ) = k (1 − 5