Neurons (nerve cells) Faculty of Science The Morris Lecar neuron - PowerPoint PPT Presentation

u n i v e r s i t y o f c o p e n h a g e n Neurons (nerve cells) Faculty of Science The Morris Lecar neuron model gives rise to the Ornstein-Uhlenbeck leaky integrate-and-fire model = ⇒ • Susanne Ditlevsen Cindy Greenwood Stochastic Models in Neuroscience Marseille 2009 January 18, 2010 Slide 1/32 u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s The model d X t = µ ( X t ) d t + σ ( X t ) d W ( t ) ; X 0 = x 0 X t : membrane potential at time t after a spike x 0 : initial voltage (the reset value following a spike) X(t) An action potential (a spike) is produced when the S membrane voltage X t exceeds a firing threshold S ( t ) = S > X (0) = x 0 After firing the process is reset to x 0 . The interspike interval T is identified with the first-passage time of the x0 threshold, T T T = inf { t > 0 : X t ≥ S } . time Slide 3/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Two commonly used Leaky Two commonly used Leaky Integrate-and-Fire neuron models (I) Integrate-and-Fire neuron models (II) The Ornstein-Uhlenbeck process: The Feller process (also CIR or square root process): � � − X t � − X t − V I � � d X t = τ + µ d t + σ d W t ; X 0 = x 0 . d( X t − V I ) = + µ d t + σ X t − V I d W t ; τ X 0 = x 0 ≥ V I . where X t : membrane potential at time t after a spike where τ : membrane time constant, reflects spontaneous V I : inhibitory reversal potential voltage decay ( > 0) and µ : characterizes constant neuronal input σ 2 2 µ ≥ σ : characterizes erratic neuronal input x 0 : initial voltage (the reset value following a spike) Slide 5/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010 Slide 6/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010 u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s OU and square-root process S 2 S 1 µτ V I time From Berg, Ditlevsen and Hounsgaard (2008) Slide 7/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s The Hodgkin-Huxley model 1.0 Hodgkin and Huxley (1952). Explains the ionic mechanisms underlying the initiation and propagation of 0.8 action potentials in the squid giant axon. Nobel Prize in Medicine in 1963. autocorrelation 0.6 0.4 Monoexponential: τ τ = = 43.1 ms 0.2 Biexponential: τ 1 = 12.1 ms; τ τ 2 = 53.8 ms 0.0 0 10 20 30 40 50 time in ms Slide 10/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010 u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s The Morris Lecar model 20 1 dV t = C ( − g Ca m ∞ ( V t )( V t − V Ca ) − g K W t ( V t − V K ) membrane voltage, V(t) − g L ( V t − V L ) + I ) dt 0 = ( α ( V t )(1 − W t ) − β ( V t ) W t ) dt dW t with the auxiliary functions given by −20 � � v − V 1 �� 1 m ∞ ( v ) = 1 + tanh 2 V 2 1 � v − V 3 � � � v − V 3 �� −40 α ( v ) = 2 φ cosh 1 + tanh 2 V 4 V 4 1 � v − V 3 � � � v − V 3 �� β ( v ) = 2 φ cosh 1 − tanh 2 V 4 V 4 0 200 400 600 800 1000 time Slide 11/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Bifurcation diagram, Morris Lecar model 0.5 normalized conductance, W(t) 0.4 0.3 0.2 ● 0.1 From Tateno and Pakdaman (2004) −40 −20 0 20 40 membrane voltage, V(t) Slide 14/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010 u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s The stochastic Morris Lecar model Where to put the noise? 1 dV t = C ( − g Ca m ∞ ( V t )( V t − V Ca ) − g K W t ( V t − V K ) − g L ( V t − V L ) + I ) dt + σ 1 ( V t , W t ) dB t dW t = ( α ( V t )(1 − W t ) − β ( V t ) W t ) dt + σ 2 ( V t , W t ) dB t Voltage noise From Tateno and Pakdaman (2004) Slide 15/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s The stochastic Morris Lecar model The stochastic Morris Lecar model W t can be interpreted as a probability and should stay Channel noise between 0 and 1. For one-dimensional diffusions dX t = b ( X t ) dt + σ ( X t ) dW t 1 dV t = C ( − g Ca m ∞ ( V t )( V t − V Ca ) − g K W t ( V t − V K ) it is easy to find conditions such that boundaries are not hit by use of the scale measure. Density: − g L ( V t − V L ) + I ) dt � x � � 2 b ( y ) s ( x ) = exp − σ 2 ( y ) dy , x ∈ ( l , r ) x ∗ dW t = ( α ( V t )(1 − W t ) − β ( V t ) W t ) dt + σ ( V t , W t ) dB t for some x ∗ ∈ ( l , r ). Density of speed measure: How should σ ( V t , W t ) look? 1 m ( x ) = σ 2 ( x ) s ( x ) , x ∈ ( l , r ) Slide 17/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010 Slide 18/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010 u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s The stochastic Morris Lecar model The stochastic Morris Lecar model If Back to business... We look at � x ∗ � r s ( y ) dy = x ∗ s ( y ) dy = ∞ = ( α ( V t )(1 − W t ) − β ( V t ) W t ) dt + σ 2 ( V t , W t ) dB t dW t l then the boundaries l and r are non-attracting. If Consider V t fixed, then for W t to stay between 0 and 1, moreover � r first of all we need the noise to go to zero when W M = m ( y ) dy < ∞ approaches the boundaries. Natural choice is a Jacobi l diffusion then X is ergodic with invariant measure µ ( x ) = m ( x ) / M . In particular, � dW t = − θ ( W t − µ ) dt + σ 2 θ W t (1 − W t ) dB t where σ 2 ≤ µ and σ 2 ≤ 1 − µ . D → µ as t → ∞ . X t The invariant distribution is a Beta-distribution with If X 0 ∼ µ , then X is stationary and X t ∼ µ for all t ≥ 0. parameters σ 2 /µ and σ 2 / (1 − µ ). Slide 19/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010 Slide 20/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s The stochastic Morris Lecar model The stochastic Morris Lecar model In our case we have We end up with the model α ( V t ) θ = α ( V t ) + β ( V t ) , µ = α ( V t ) + β ( V t ) 1 Requirements translates to dV t = C ( − g Ca m ∞ ( V t )( V t − V Ca ) − g K W t ( V t − V K ) α ( V t ) σ 2 ≤ − g L ( V t − V L ) + I ) dt α ( V t ) + β ( V t ) β ( V t ) σ 2 ≤ α ( V t ) + β ( V t ) dW t = ( α ( V t )(1 − W t ) − β ( V t ) W t ) dt Fulfilled if � + σ 2 α ( V t ) β ( V t ) W t (1 − W t ) dB t α ( V t ) β ( V t ) σ 2 ≤ α ( V t ) + β ( V t ) where σ ≤ 1. Now V t is not fixed, but still okay... when 0 < α ( V t ) , β ( V t ) < 1. Slide 21/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010 Slide 22/32— Susanne Ditlevsen — The Morris Lecar and Ornstein-Uhlenbeck LIF neuron model — January 18, 2010 0.5 0.5 0.5 normalized conductance, W(t) normalized conductance, W(t) normalized conductance, W(t) 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 ● ● 0.1 0.1 0.2 −40 −20 0 20 40 −40 −20 0 20 40 membrane voltage, V(t) membrane voltage, V(t) ● 0.1 −40 −20 0 20 40 membrane voltage, V(t)