From experiments to models R.C. Lambert Neuronal networks and - PowerPoint PPT Presentation

From experiments to models R.C. Lambert Neuronal networks and physiopathological rhythms Université Pierre et Marie Curie- Neuroscience Paris Seine (NPS) CNRS UMR8246, INSERM U1130, UPMC UM119

How do I model a cortical neuron ? cortical pyramidal neuron current clamp mode Ie Vm Vm Ie

How do I model a cortical neuron ? cortical pyramidal neuron current clamp mode Ie Vm decreasing frequency (adaptation) spikes amplitude of the depolarization slow depolarization resting Vm (-70mV)

Resting membrane potential V = EeqK + - + K+ - K+ K+ + K+ - + - + - + - + A- A- A- A- - + - + o o i i dynamic equilibrium Assume channel only conducts one species of ion (e.g. K + ) • Consider energy change in transferring one mole of ions across the membrane: • electrical potential change = zFEeq chemical potential change = RT ln([ion] i / [ion] o ) Nernst equation : Eeq = - (RT / zF) ln([ion] i / [ion] o )

Resting membrane potential In an electrolyte solution, the concentrations of all the ions are such that the solution is electrically neutral. V = EeqK + - + - K+ + K+ - + - + - + - + A- A- - + - + o i

Resting membrane potential In an electrolyte solution, the concentrations of all the ions are such that the solution is electrically neutral. V = EeqK + - + - K+ + K+ - + - + - + - + A- A- - + - + o i 10 Å + - 10 Å

Resting membrane potential Cm V = EeqK + - + - K+ + K+ - + o i - + - + - + A- A- - + « A capacitor consists of two conductors separated by a - + non-conductive region » (Wikipedia :-)) o i • The capacitor behaves as a charge storage device • The charge accumulation (Q) in the capacitor plates is not instantaneous. Potential across the capacitor exponentially rises Q = C . E eq − > I c = dQ dt = C . dV 10 Å dt + The value of the capacitance C (in F) is proportional to the - nature of the dielectric (cste), its thickness (cste) and the area of the « plates » (size of the cell). 10 Å Cm = cmA with cm (specific capacitance) = 10nF/mm2 and A in mm2 (0.01 to 0.1 mm2)

Resting membrane potential Cm V = EeqK + - + - K+ + K+ - + o i - + - + - + A- A- - GK + + - + o i EeqK + • The K+ conductance is not voltage dependent • The K+ current = 0 when V=EeqK+ I K = G K . ( V − Eeq K )

Resting membrane potential • other intracellular anions (non-permeant except bicarbonate) : proteins, metabolites, phosphates, bicarbonate • gradients maintained by pumps (Na/K-ATPase: electrogenic 3Na/2K) Na + ; Ca 2+ K + • Veq near -70mV ➢ mostly K+ conductance (anion fluxes = 0) 2+ A T P Ca + Na Cl - ATP + K

Resting membrane potential Cm V inv = G Na . Eeq Na + G K . Eeq K G Na + G K o i GNa + - + EeqNa + - K+ + K+ - + Na+ Na+ - + - + A- A- - GK + + EeqK + - + o i equilibrium Cm o i G G leak = 1 E leak = V inv R m

How do I model a cortical neuron ? cortical pyramidal neuron current clamp mode Ie Vm decreasing frequency (adaptation) spikes Cm amplitude of the depolarization o i slow depolarization resting Vm (-70mV) G Eleak Gleak

Passive response to input Cm dVm = dQ dt = I c dt Vm Cm Ic 0 = Ic + I leak = Cm dVm + G leak ( Vm − E leak ) i o dt I leak At rest, Vm = cst Vm=Eleak G Eleak Gleak I leak = G leak ( Vm − E leak ) rem : by convention, Iion >0 means output of cations. membrane current is defined as postive-outward

Passive response to input Cm dVm = dQ dt = I c dt Vm Cm Ic Ie i o I leak (Kirch-hoff’s law : sum of all the currents entering a point in a circuit must be 0) − Ie + Ic + I leak = 0 G Eleak Gleak − Ie + Ic + I leak = − Ie + Cm dVm + G leak ( Vm − E leak ) = 0 dt I leak = G leak ( Vm − E leak ) Cm dVm = Ie − G leak ( Vm − E leak ) dt rem : by convention, current that enters the neuron through an electrode is defined as positive-inward

Passive response to input Cm dVm = Ie − G leak ( Vm − E leak ) dt Rm . Cm . dVm = Rm . Ie − ( Vm − E leak ) dt Rm . Cm . dVm = ( Vm ∞ − E leak ) − ( Vm − E leak ) dt Rm . Cm . dVm = Vm ∞ − Vm dt − t Δ Vm Vm = Vm ∞ . [1 − exp ( Rm . Cm )] Vm = Vm ∞ . [1 − exp ( − t )] ; with τ m = Rm . Cm (time cste) τ m Rm and Cm define how the membrane potential of the cell changes in response to ion fluxes (synaptic activity, electrode, voltage dependent conductances …) both in term of amplitude ( ∆ Vm= Rm.Ie) and kinetics ( 𝛖 m= Rm.Cm)

How do I model a cortical neuron ? cortical pyramidal neuron current clamp mode Ie Vm decreasing frequency (adaptation) spikes Cm amplitude of the depolarization o i slow depolarization resting Vm (-70mV) G Eleak Gleak

How do I model a cortical neuron ? The simplest neuron model : Leaky Integrate and Fire Cm dVm = Ie − G leak ( Vm − E leak ) dt • only passive properties when Vm reaches a threshold value V th (-50, -55mV) the neuron fires an AP and Vm is reset to V reset (E leak ) • if Ie=0, Vm relaxes exponentially to E leak with time constant 𝝊 •

How do I model a cortical neuron ? cortical pyramidal neuron current clamp mode Ie Vm decreasing frequency (adaptation) spikes amplitude of the depolarization slow depolarization resting Vm (-70mV)

How do I model a cortical neuron ? More complex Integrate and Fire Cm dVm = Ie − G leak ( Vm − E leak ) − G adapt ( Vm − E adapt ) dt when Vm reaches a threshold value V th (-50, -55mV) : • • the neuron fires an AP Vm is reset to V reset (E leak ) • G adapt is increased by an amount Δ G adapt -> during repetitive firing I adapt progressively • increases, slowing spike generation decreasing frequency (adaptation) with G adapt cortical neuron see for example : Gerstner, W. (2008). Spike-response model. Scholarpedia , 3 (12), 1343. http://doi.org/10.4249/scholarpedia.1343

How do I model a cortical neuron ? Adding voltage-dependent conductances G Na + E eq Na + o i G K + E eq K + Hodgkin & Huxley 1939 Hodgkin & Huxley 1952 E leak G leak Cm . dVm = Ie − G leak . ( Vm − E leak ) − I Na ( t , Vm ) − I K ( t , Vm ) dt Cm . dVm = Ie − G leak . ( Vm − E leak ) − G Na ( t , Vm ) . ( Vm − E Na ) − G K ( t , Vm ) . ( Vm − E K ) dt

the theory Bezanilla 2000 Armstrong 2003 I K = G K ( t , Vm ) . ( Vm − E K ) • probability of one « gate » to be open : n • probability of the 4 « gates » to be open : n 4 • maximal K+ conductance : G K • K+ conductance of the cell : G K n 4 I K = G K . n 4 ( t , Vm ) . ( Vm − E K )

the theory α ( Vm ) • probability of one « gate » to be open : n 1 - n( t,Vm ) n( t,Vm ) • probability of the 4 « gates » to be open : n 4 • maximal K+ conductance : G K open closed β ( Vm ) • K+ conductance of the cell : G K n 4 probability probability at a given Vm : n ( t + δ t ) = n ( t ) + α . (1 − n ( t )) . δ t − β . n ( t ) . δ t n ( t + δ t ) − n ( t ) = α . (1 − n ( t )) − β . n ( t ) δ t dn dt = α − ( α + β ) . n α + β . dn 1 α dt = α + β − n 1 α τ . dn with τ = α + β and n ∞ = dt = n ∞ − n α + β − t τ ) n ( t ) = n 0 − ( n 0 − n ∞ ) . (1 − e I K = G K . n 4 ( t , Vm ) . ( Vm − E K ) ; the kinetics of the current at Vm is defined by τ and n ∞

experiments I K = G K . n 4 ( t , Vm ) . ( Vm − E K ) ; the kinetics of the current at Vm is defined by τ and n ∞ Recording the current evoked at different Vm : • control Vm : voltage-clamp (impossible in vivo , non perfect in vitro , achievable for recombinant channels) • isolate the current : pharmacology, recording solutions I K ∞ = G K . n 4 ∞ ( Vm ) . ( Vm − E K ) I K ∞ V m +40 I K ∞ n 4 ∞ ( Vm ) = -60 G K . ( Vm − E K ) -80mV

experiments I K = G K . n 4 ( t , Vm ) . ( Vm − E K ) ; the kinetics of the current at Vm is defined by τ and n ∞ I K ∞ = G K . n 4 ∞ ( Vm ) . ( Vm − E K ) n 4 ∞ V m +40 I K ∞ 1 n 4 ∞ ( Vm ) = G K . ( Vm − E K ) = -60 -80mV V 0.5 − Vm 1 + e k Boltzman equation

experiments I K = G K . n 4 ( t , Vm ) . ( Vm − E K ) ; the kinetics of the current at Vm is defined by τ and n ∞ − t τ ) n ( t ) = n 0 − ( n 0 − n ∞ ) . (1 − e at each fixed Vm : τ )] 4 . ( Vm − E K ) − t I K = G K . [ n 0 − ( n 0 − n ∞ ) . (1 − e − t τ )] 4 I K = cste . (1 − e 𝝊 deactivation 𝝊 activation 𝝊 +40 -60 -80mV V m

the theory voltage dependent conductances have potentially 3 main states : Closed - Open - Inactivated α m ( Vm ) 1 - m m β m ( Vm ) G Na + E eq Na + α h ( Vm ) G K + E eq K + 1 - h h -5 β h ( Vm ) -80 -110mV E leak G leak

experiments voltage dependent conductances have potentially 3 main states : Closed - Open - Inactivated G Na + E eq Na + G K + E eq K + m 3 ∞ h ∞ E leak G leak -5 -30 -130mV

experiments voltage dependent conductances have potentially 3 main states : Closed - Open - Inactivated G Na + E eq Na + inactivation kinetics G K + E eq K + E leak G leak 𝝊 (Vm) deinactivation kinetics -5 -30 -130mV

Single compartment HH model G Na + E eq Na + G K + E eq K + E leak G leak Cm . dVm = Ie − G leak . ( Vm − E leak ) − G K . n 4 ( t , Vm ) . ( Vm − E K ) − G Na . m 3 ( t , Vm ) . h ( t , Vm ) . ( Vm − E Na ) dt … add other ionic conductances to include more complex behaviors such as adaptation