The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook Modelling neuronal membranes with Piecewise Deterministic Processes Martin Riedler work with Evelyn Buckwar Heriot–Watt University, Edinburgh, UK Stochastic Models in Neuroscience, CIRM Marseille, Jan 19 th , 2010 1 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook Outline The Modelling Problem 1 Piecewise Deterministic Processes (PDPs) in finite dimensions 2 Spatial Dynamics – PDPs in infinite dimensions 3 Outlook 4 2 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook Scale levels and modelling approaches I. atomic level II. protein level III. cellular level charged ions involved in large numbers; one single channel consists of hun- behaviour of transients important is their distribution dreds of amino acids, hence over length scales of 1 µm close to the membrane and their flux channel ≫ ion and possibly only to 1 m ; fitting of a deter- across the membrane; flux rate is a few channels involved ∼ low ministic model to the average 10 3 − 10 6 ions/ms per open channel; channel density or small fibres; emergent behaviour; discrete continuous discrete approximation modelling modelling microscopic description hybrid stochastic models macroscopic continuous discrete particle modelling system continuous noise approximation deterministic limit continuous stochastic models macroscopic model SDE/SPDE ODE/PDE models models "adding" noise 3 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook Motivation for and aim of our work Motivation several hybrid algorithms proposed that reproduce noisy excitable behaviour for neuron models [Skaugen/Walloe 79, Clay/DeFelice 83, Chow/White 96, etc.] ; ”noise = channel noise” in context of neuronal modelling, . . . . . . hardly any theory or rigorous analysis of hybrid processes and their numerical approximation . . . hardly any analysis of continuous stochastic approximations, their appropriateness and quality Our approach and aims use PDPs which provide accurate mathematical description hybrid model (in the space-clamped setting) well known class of stochastic processes with existing theory extend PDP theory to include spatial dynamics develop tools for theoretically analysing hybrid algorithms 4 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook Building blocks of a hybrid stochastic model Single channel model – continuous-time Markov chain → waiting time distr., e.g. Example: (simplified) Na -channel P [ τ > t ] = e − ( a m ( v )+ a h ( v )) t a m ( v ) → post-jump value distr., e.g. m 0 h 0 ⇋ m 1 h 0 b m ( v ) a m ( v ) a h ( v ) ⇃ ↾ b h ( v ) a h ( v ) ⇃ ↾ b h ( v ) p m 1 h 0 = a m ( v ) + a h ( v ) , a m ( v ) m 0 h 1 ⇋ m 1 h 1 a h ( v ) b m ( v ) p m 0 h 1 = a m ( v ) + a h ( v ) . → transmembrane potential constant over time 5 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook Building blocks of a hybrid stochastic model Single channel model – continuous-time Markov chain → waiting time distr., e.g. Example: (simplified) Na -channel P [ τ > t ] = e − ( a m ( v )+ a h ( v )) t a m ( v ) → post-jump value distr., e.g. m 0 h 0 ⇋ m 1 h 0 b m ( v ) a m ( v ) a h ( v ) ⇃ ↾ b h ( v ) a h ( v ) ⇃ ↾ b h ( v ) p m 1 h 0 = a m ( v ) + a h ( v ) , a m ( v ) m 0 h 1 ⇋ m 1 h 1 a h ( v ) b m ( v ) p m 0 h 1 = a m ( v ) + a h ( v ) . → transmembrane potential constant over time (Space-clamped) membrane model – ordinary differential equation m � C ˙ v = g i ( v ) ( E i − v ) i =1 conductances g i ∝ fraction of open channels 5 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook combining these building blocks: transmembrane potential is not constant → single channel models are not Markovian anymore the correct waiting distribution for a channel in time-varying potential s �→ v ( s ) is, e.g., Example cont. R t P [ τ > t ] = e − 0 a m ( v ( s ))+ a h ( v ( s ))d s 6 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook combining these building blocks: transmembrane potential is not constant → single channel models are not Markovian anymore the correct waiting distribution for a channel in time-varying potential s �→ v ( s ) is, e.g., Example cont. R t P [ τ > t ] = e − 0 a m ( v ( s ))+ a h ( v ( s ))d s → using this waiting time distribution [Clay/DeFelice, 83] propose an approximation algorithm for a space-clamped membrane → no analysis of the algorithm; particularly, a strong Markov property of the hybrid process ”(voltage, channel state)” needed for the algorithm to be correct 6 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook (finite-dimensional) Piecewise Deterministic Process - Definition Davis, Markov Models and Optimization, 1994 ag process ( y ( t )) t ≥ 0 , where y ( t ) = ( x ( t ) , r ( t )) ∈ R d × R , with |R| < ∞ . c` adl` continuous, deterministic dynamics – cont. component x ( t ) a family of ODEs x = g r ( x ) ˙ with solutions t �→ φ r ( t ; x 0 ) for all x (0) = x 0 ∈ R d and every r ∈ R discrete, stochastic dynamics – pwc. component r ( t ) jump rate λ : R d × R → R + , s. t. for the jump times 0 < τ 1 < τ 2 < . . . of r ( t ) with τ k < T Z t “ ” ` ´ P [ τ k +1 − T > t | r ( T ) = r, x ( T ) = x ] = exp − λ φ r ( s ; x ) , r d s 0 transition measure Q : R d × R → P ( R ) for post-jump values P [ r ( τ k +1 ) = ¯ r | r ( τ k +1 − ) = r, x ( T ) = x ] = Q ( φ r ( τ k +1 − T ; x ) , r )( { ¯ r } ) 7 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook Properties under certain assumptions ... strong, homogeneous Markov process theoretically, an exact simulation algorithm exist; practically, for each inter-jump interval ˙ x = g r ( x ) and, simultaneously, to obtain the next jump time � t � � − log U = φ r ( s ; x ) , r d s, λ 0 with U ∼ U (0 , 1) , has to be solved numerically. generator A of the Markov process y ( t ) and its domain D ( A ) exactly defined, i.e., for f ∈ D ( A ) � A f ( x, r ) = f x ( x, r ) g r ( x ) + λ ( x, r ) [ f ( x, p ) − f ( x, r )] Q ( x, r )(d p ) R → starting point for the approximation with continuous processes generalisation to include spatial dynamics [Buckwar, R., in prep.] 8 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook Example: space-clamped, leaky membrane with N channels a m ( v ) v = ¯ ˙ g Na r m 1 h 1 ( E Na − v ) − g L v m 0 h 0 ⇋ m 1 h 0 | {z } b m ( v ) = g r ( v ) a h ( v ) ⇃ ↾ b h ( v ) a h ( v ) ⇃ ↾ b h ( v ) a m ( v ) r = ( r m 0 h 0 , r m 1 h 0 , r m 0 h 1 , r m 1 h 1 ) , m 0 h 1 ⇋ m 1 h 1 b m ( v ) r i = # channels in state i 9 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook Example: space-clamped, leaky membrane with N channels a m ( v ) v = ¯ ˙ g Na r m 1 h 1 ( E Na − v ) − g L v m 0 h 0 ⇋ m 1 h 0 | {z } b m ( v ) = g r ( v ) a h ( v ) ⇃ ↾ b h ( v ) a h ( v ) ⇃ ↾ b h ( v ) a m ( v ) r = ( r m 0 h 0 , r m 1 h 0 , r m 0 h 1 , r m 1 h 1 ) , m 0 h 1 ⇋ m 1 h 1 b m ( v ) r i = # channels in state i � r ∈ R 4 : r i ∈ { 0 : 1 : N } and � � R = r i = N i jump rate � � λ ( v, r ) = r · a m ( v )+ a h ( v ) , b m ( v )+ a h ( v ) , a m ( v )+ b h ( v ) , b m ( v )+ b h ( v ) transition probabilities, e.g. a m ( v ) P [ r → r + ( − 1 , 1 , 0 , 0)] = r m 0 h 0 λ ( v, r ) 9 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook # channels =1000 # channels =100 # channels = 500 150 150 150 monophasic stimulus 100 100 100 50 50 50 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 150 150 150 preconditioned stimulus 100 100 100 50 50 50 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 10 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook infinite-dimensional Piecewise Deterministic Process - Definition [Buckwar, R., Preprint in prep.] separable, real Hilbert space E continuously and densely embedded in and a Borel subset of another separable, real Hilbert space H ; further H ⊂ E ∗ ; c` adl` ag process ( y ( t )) t ≥ 0 , where y ( t ) = ( u ( t ) , r ( t )) ∈ E × R , with |R| < ∞ . continuous, deterministic dynamics – cont. component u ( t ) a family of abstract ODEs u = L r u + G r ( u ) ˙ ◦ L r : E → E ∗ linear operators ◦ G r : E → E ∗ nonlinear mappings with solutions t �→ ψ r ( t ; u 0 ) for all u (0) = u 0 ∈ E and every r ∈ R discrete, stochastic dynamics – pwc. component r ( t ) jump rate Λ : E × R → R + , transition measure Q : E × R → P ( R ) 11 / 16
The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook Properties under certain assumptions ... strong, homogeneous Markov process if the strong (Fr´ echet-) derivative of f ∈ D ( A ) w.r.t. the u component exists, then the extended generator of y is given by � A f ( u, r ) = � L r u + G r ( u ) , f u ( u, r ) � +Λ r ( u ) [ f ( u, p ) − f ( u, r )] Q r (d p ; u ) R with �· , ·� duality pairing of E and f u the unique element of E s.t. d f d x ( x, r ) ◦ y = ( y, f x ( x, r )) E ∀ y ∈ E d u ( u, r ) ∈ E ∗ Fr´ where d f echet derivative of f w.r.t. u at ( u, r ) . → in case of higher (spatial) regularity we can use the inner product on H instead of �· , ·� 12 / 16
Recommend
More recommend
