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Lecture 7: Stochastic models of channels, synapses References: Dayan & Abbott, Sects 5.7, 5.8 Gerstner & Kistler, Sect 2.4 C Koch, Biophysics of Computation Chs 4,8 (13) A Destexhe, Z Mainen & T J Sejnowski, Ch 1 in Methods in Neuronal Modeling, 2nd ed, C Koch and I Segev, eds (MIT Press) Stochastic models of channels Single channels are stochastic, described by kinetic equations for probabilities of being in different states Stochastic models of channels Single channels are stochastic, described by kinetic equations for probabilities of being in different states Example: the HH K channel: HH K channel Kinetic equations: dp1 n p2 4 n p1 dt dp2 4 n p1 2 n p3 ( n 3 n ) p2 dt dp3 3 n p2 3 n p4 (2 n 2 n ) p3 dt dp4 2 n p3 4 n p5 (3 n n ) p4 dt dp5 n p4 4 n p5 dt HH K channel Kinetic equations: dp1 n p2 4 n p1 dt dp2 4 n p1 2 n p3 ( n 3 n ) p2 dt dp3 3 n p2 3 n p4 (2 n 2 n ) p3 dt dp4 2 n p3 4 n p5 (3 n n ) p4 dt dp5 n p4 4 n p5 dt Open probability: n = p5 HH Na Channel HH Na Channel HH Na Channel But in this picture, inactivation only when activation gate is open: Na channel: Patlak model Na channel: Patlak model V-independent k1, k2, k3 Fits fast data a bit better than stochastic HH model Synapses Synapses Conductances gated by presynaptic activity: Synapses Conductances gated by presynaptic activity: I syn (t ) g syn (t )(V (t ) Vrev ) Synapses Conductances gated by presynaptic activity: I syn (t ) g syn (t )(V (t ) Vrev ) gs gs P Synapses Conductances gated by presynaptic activity: I syn (t ) g syn (t )(V (t ) Vrev ) gs gs P P Ps Prel Synapses Conductances gated by presynaptic activity: I syn (t ) g syn (t )(V (t ) Vrev ) gs gs P P Ps Prel ~ deterministic (many channels) on postsynaptic side, stochastic on presynaptic side Synapses Conductances gated by presynaptic activity: I syn (t ) g syn (t )(V (t ) Vrev ) gs gs P P Ps Prel ~ deterministic (many channels) on postsynaptic side, stochastic on presynaptic side Receptors: ionotropic and metabotropic Synapses Conductances gated by presynaptic activity: I syn (t ) g syn (t )(V (t ) Vrev ) gs gs P P Ps Prel ~ deterministic (many channels) on postsynaptic side, stochastic on presynaptic side Receptors: ionotropic and metabotropic Synapses Conductances gated by presynaptic activity: I syn (t ) g syn (t )(V (t ) Vrev ) gs gs P P Ps Prel ~ deterministic (many channels) on postsynaptic side, stochastic on presynaptic side Receptors: ionotropic and metabotropic Transmitters and Receptors Main transmitters: Transmitters and Receptors Main transmitters: glutamate (excitatory) Transmitters and Receptors Main transmitters: glutamate (excitatory) GABA (g-aminobutyric acid, inhibitory) Transmitters and Receptors Main transmitters: glutamate (excitatory) GABA (g-aminobutyric acid, inhibitory) ACh (neuromuscular junction) Transmitters and Receptors Main transmitters: glutamate (excitatory) GABA (g-aminobutyric acid, inhibitory) ACh (neuromuscular junction) Noradrenaline (modulatory) Transmitters and Receptors Main transmitters: glutamate (excitatory) GABA (g-aminobutyric acid, inhibitory) ACh (neuromuscular junction) Noradrenaline (modulatory) Receptor types (named after pharmacological agonists): Transmitters and Receptors Main transmitters: glutamate (excitatory) GABA (g-aminobutyric acid, inhibitory) ACh (neuromuscular junction) Noradrenaline (modulatory) Receptor types (named after pharmacological agonists): Glutamate receptors (both ionotropic) : Transmitters and Receptors Main transmitters: glutamate (excitatory) GABA (g-aminobutyric acid, inhibitory) ACh (neuromuscular junction) Noradrenaline (modulatory) Receptor types (named after pharmacological agonists): Glutamate receptors (both ionotropic) : AMPA (Na, K) NMDA (Na, K, Ca) Transmitters and Receptors Main transmitters: glutamate (excitatory) GABA (g-aminobutyric acid, inhibitory) ACh (neuromuscular junction) Noradrenaline (modulatory) Receptor types (named after pharmacological agonists): Glutamate receptors (both ionotropic) : AMPA (Na, K) NMDA (Na, K, Ca) GABA receptors Transmitters and Receptors Main transmitters: glutamate (excitatory) GABA (g-aminobutyric acid, inhibitory) ACh (neuromuscular junction) Noradrenaline (modulatory) Receptor types (named after pharmacological agonists): Glutamate receptors (both ionotropic) : AMPA (Na, K) NMDA (Na, K, Ca) GABA receptors GABAA (ionotropic, Cl) GABAB (metabotropic, K) Transmitters and Receptors Main transmitters: glutamate (excitatory) GABA (g-aminobutyric acid, inhibitory) ACh (neuromuscular junction) Noradrenaline (modulatory) Receptor types (named after pharmacological agonists): Glutamate receptors (both ionotropic) : AMPA (Na, K) NMDA (Na, K, Ca) GABA receptors GABAA (ionotropic, Cl) GABAB (metabotropic, K) Ach receptors: Transmitters and Receptors Main transmitters: glutamate (excitatory) GABA (g-aminobutyric acid, inhibitory) ACh (neuromuscular junction) Noradrenaline (modulatory) Receptor types (named after pharmacological agonists): Glutamate receptors (both ionotropic) : AMPA (Na, K) NMDA (Na, K, Ca) GABA receptors GABAA (ionotropic, Cl) GABAB (metabotropic, K) Ach receptors: nicotinic (ionotropic) muscarinic (metabotropic) Postsynaptic conductance (AMPA receptor) Kinetic equation: Postsynaptic conductance (AMPA receptor) Kinetic equation: dPs s (1 Ps ) s Ps dt Postsynaptic conductance (AMPA receptor) Kinetic equation: dPs s (1 Ps ) s Ps dt Transmitter: s constant for a short time, s >> s Postsynaptic conductance (AMPA receptor) Kinetic equation: dPs s (1 Ps ) s Ps dt Transmitter: s constant for a short time, s >> s Ps (t ) 1 ( Ps (0) 1) exp( s t ), 0t T Postsynaptic conductance (AMPA receptor) Kinetic equation: dPs s (1 Ps ) s Ps dt Transmitter: s constant for a short time, s >> s Ps (t ) 1 ( Ps (0) 1) exp( s t ), 0t T Then =0, decay: Ps (t ) Ps (T ) exp( s (t T )), t T Postsynaptic conductance (AMPA receptor) Kinetic equation: dPs s (1 Ps ) s Ps dt Transmitter: s constant for a short time, s >> s Ps (t ) 1 ( Ps (0) 1) exp( s t ), 0t T Then =0, decay: Ps (t ) Ps (T ) exp( s (t T )), t T Postsynaptic conductance (AMPA receptor) Kinetic equation: dPs s (1 Ps ) s Ps dt Transmitter: s constant for a short time, s >> s Ps (t ) 1 ( Ps (0) 1) exp( s t ), 0t T Then =0, decay: Ps (t ) Ps (T ) exp( s (t T )), t T s = 0.93/ms s = 0.19/ms Other receptors excitatory inhibitory Other receptors excitatory inhibitory commonly fit by Ps 1 [exp( t / 1 ) exp( t / 2 )] 1 2 1 2 Other receptors excitatory inhibitory commonly fit by Ps 1 [exp( t / 1 ) exp( t / 2 )] 1 2 limit 1 2 : Ps t 2 exp( t / ) 1 2 Other receptors excitatory inhibitory commonly fit by Ps 1 [exp( t / 1 ) exp( t / 2 )] 1 2 limit 1 2 : Ps t 2 exp( t / ) 1 2 “-function” NMDA receptors Conductance is voltage-dependent (raising voltage knocks out Mg ions that block channel at low V) NMDA receptors Conductance is voltage-dependent (raising voltage knocks out Mg ions that block channel at low V) I NMDA g NMDA Ps (V VNMDA ) NMDA receptors Conductance is voltage-dependent (raising voltage knocks out Mg ions that block channel at low V) I NMDA g NMDA Ps (V VNMDA ) g NMDA 0 g NMDA [Mg 2 ] 1 exp( V / 16.13 mV) 3.57 mM NMDA receptors Conductance is voltage-dependent (raising voltage knocks out Mg ions that block channel at low V) I NMDA g NMDA Ps (V VNMDA ) g NMDA 0 g NMDA [Mg 2 ] 1 exp( V / 16.13 mV) 3.57 mM NMDA receptors Conductance is voltage-dependent (raising voltage knocks out Mg ions that block channel at low V) I NMDA g NMDA Ps (V VNMDA ) g NMDA Opening requires both preand postsynaptic depolarization: Coincidence detector (important for learning) 0 g NMDA [Mg 2 ] 1 exp( V / 16.13 mV) 3.57 mM GABAB receptor kinetics Simplest model for a metabotropic receptor: GABAB receptor kinetics Simplest model for a metabotropic receptor: Transmitter binding activates receptor: GABAB receptor kinetics Simplest model for a metabotropic receptor: Transmitter binding activates receptor: dr r [T ](1 r ) r r dt GABAB receptor kinetics Simplest model for a metabotropic receptor: Transmitter binding activates receptor: dr r [T ](1 r ) r r dt Active receptor activates second messenger: GABAB receptor kinetics Simplest model for a metabotropic receptor: Transmitter binding activates receptor: dr r [T ](1 r ) r r dt Active receptor activates second messenger: ds kr s s dt GABAB receptor kinetics Simplest model for a metabotropic receptor: Transmitter binding activates receptor: dr r [T ](1 r ) r r dt Active receptor activates second messenger: ds kr s s dt Cooperative binding of second messenger to K channel opens it for current: GABAB receptor kinetics Simplest model for a metabotropic receptor: Transmitter binding activates receptor: dr r [T ](1 r ) r r dt Active receptor activates second messenger: ds kr s s dt Cooperative binding of second messenger to K channel opens it for current: I GABAB g GABAB s4 (V VK ) s4 K Presynaptic kinetics: depression and facilitation Presynaptic kinetics: depression and facilitation depression (exc->exc synapses) Presynaptic kinetics: depression and facilitation depression (exc->exc synapses) facilitation (exc->inh synapses) Synaptic depression Dynamics of Prel controlled by depletion of synaptic vesicles: Synaptic depression Dynamics of Prel controlled by depletion of synaptic vesicles: dPrel 1 Prel U (t t s ) Prel dt r Synaptic depression Dynamics of Prel controlled by depletion of synaptic vesicles: dPrel 1 Prel U (t t s ) Prel dt r For presynaptic rate r(t), Synaptic depression Dynamics of Prel controlled by depletion of synaptic vesicles: dPrel 1 Prel U (t t s ) Prel dt r For presynaptic rate r(t), dPrel 1 Prel Ur(t ) Prel dt r Synaptic depression Dynamics of Prel controlled by depletion of synaptic vesicles: dPrel 1 Prel U (t t s ) Prel dt r For presynaptic rate r(t), dPrel 1 Prel Ur(t ) Prel dt r For stationary rate, stationary solution is Prel0 1 1 Ur r Synaptic depression Dynamics of Prel controlled by depletion of synaptic vesicles: dPrel 1 Prel U (t t s ) Prel dt r For presynaptic rate r(t), dPrel 1 Prel Ur(t ) Prel dt r For stationary rate, stationary solution is Prel0 1 1 Ur r Response to change in presynaptic rate expand: Prel Prel0 Prel Response to change in presynaptic rate expand: Prel Prel0 Prel 1 d Prel Urfinal Prel dt r Prel r Prel0, final Response to change in presynaptic rate expand: Prel Prel0 Prel 1 d Prel Urfinal Prel dt r Prel r Prel0, final Response to change in presynaptic rate expand: Prel Prel0 Prel 1 d Prel Urfinal Prel dt r Prel r Prel0, final Responds to change in input, not much to absolute level Synaptic facilitation Prel = P(vesicle) P(release|vesicle) Synaptic facilitation Prel = P(vesicle) P(release|vesicle) x y Synaptic facilitation Prel = P(vesicle) P(release|vesicle) x y Dynamics of x: depression (vesicle depletion) Synaptic facilitation Prel = P(vesicle) P(release|vesicle) x y Dynamics of x: depression (vesicle depletion) Dynamics of y: facilitation (need Ca influx to make release possible) Synaptic facilitation Prel = P(vesicle) P(release|vesicle) x y Dynamics of x: depression (vesicle depletion) Dynamics of y: facilitation (need Ca influx to make release possible) dy y0 y f (1 y) (t t s ) dt f Synaptic facilitation Prel = P(vesicle) P(release|vesicle) x y Dynamics of x: depression (vesicle depletion) Dynamics of y: facilitation (need Ca influx to make release possible) dy y0 y f (1 y) (t t s ) dt f For stationary rate: y y0 fr f 1 fr f Synaptic facilitation Prel = P(vesicle) P(release|vesicle) x y Dynamics of x: depression (vesicle depletion) Dynamics of y: facilitation (need Ca influx to make release possible) dy y0 y f (1 y) (t t s ) dt f For stationary rate: y y0 fr f 1 fr f Combined model (Markram-Tsodyks) Combined model (Markram-Tsodyks) Facilitation as before: Combined model (Markram-Tsodyks) Facilitation as before: dy y0 y f (1 y) (t t s ) dt f Combined model (Markram-Tsodyks) Facilitation as before: dy y0 y f (1 y) (t t s ) dt f Depression is proportional to Prob(release|vesicle) after spike: Combined model (Markram-Tsodyks) Facilitation as before: dy y0 y f (1 y) (t t s ) dt f Depression is proportional to Prob(release|vesicle) after spike: dx 1 x [ y f (1 y )] (t t s ) x dt r Combined model (Markram-Tsodyks) Facilitation as before: dy y0 y f (1 y) (t t s ) dt f Depression is proportional to Prob(release|vesicle) after spike: dx 1 x [ y f (1 y )] (t t s ) x dt r With presynaptic rate r(t): Combined model (Markram-Tsodyks) Facilitation as before: dy y0 y f (1 y) (t t s ) dt f Depression is proportional to Prob(release|vesicle) after spike: dx 1 x [ y f (1 y )] (t t s ) x dt r With presynaptic rate r(t): dy y0 y f (1 y)r (t ) dt f Combined model (Markram-Tsodyks) Facilitation as before: dy y0 y f (1 y) (t t s ) dt f Depression is proportional to Prob(release|vesicle) after spike: dx 1 x [ y f (1 y )] (t t s ) x dt r With presynaptic rate r(t): dy y0 y f (1 y)r (t ) dt f dx 1 x [ y f (1 y )]r (t ) x dt r