A Calcium-Based Model Of Spike- Timing Dependent Plasticity

Transcription

1 A Calcium-Based Of Spike- Timing Dependent Plasticity By Gaetan Vignoud 1,2, Jonathan Touboul 2 and Laurent Venance 2 1 Ecole Normale Supérieure, 2 Collège de France (CIRB)

2 Spike-Timing Dependent Plasticity and memory - Learning and memory associated with synaptic plasticity - Hebb s postulate - Importance of the timing and the order (pre-post or post-pre) - t = t %&'( t %+

3 Spike-Timing Dependent Plasticity and memory - Learning and memory associated with synaptic plasticity - Hebb s postulate - Importance of the timing and the order (pre-post or post-pre) Pre Post - t = t %&'( t %+ [Feldman, 2012, Neuron]

4 Anti-Hebbian STPD in corticostriatal synapses STDP = t N %34456' Freq a b N=100 pairings Stimulation Cortex Striatum Recording & stimulation Normalized EPSC (%) x (-30< t<0ms) (n=10) 100x (0< t<+30ms) (n=7) pa 25 ms Time (min) [Cui et al, 2015, J. Physiol.]

5 Anti-Hebbian STPD in corticostriatal synapses STDP = t N %34456' Freq Pre-post pairings lead to LTD - Post-pre pairings lead to LTP Anti-Hebbian learning rule Δt STDP (ms) 50 0 Normalized EPSC (%) [Cui et al, 2015, J. Physiol.]

6 Importance of the number of pairings STDP = t N Pairings Freq Normalized EPSC (%) < t<0ms (post-pre pairings) 0< t<+30ms (pre-post pairings) Number of pairings (N) [Cui et al, 2015, J. Physiol.]

7 Importance of the number of pairings STDP = t N Pairings Freq Normalized EPSC (%) NMDA dependent -30< t<0ms (post-pre pairings) 0< t<+30ms (pre-post pairings) Number of pairings (N) [Cui et al, 2015, J. Physiol.]

8 Importance of the number of pairings STDP = t N Pairings Freq Normalized EPSC (%) Endocannabinoids dependent -30< t<0ms (post-pre pairings) 0< t<+30ms (pre-post pairings) Number of pairings (N) [Cui et al, 2015, J. Physiol.]

9 Different models of STDP STDP = t N Pairings Freq - Several biophysical models explain STDP curves as [Graupner and Brunel, 2012, PNAS] based on the postsynaptic calcium trace - Anti-Hebbian STDP curve Change in synaptic efficacy t (s) [Graupner and Brunel, 2012, PNAS]

10 Different models of STDP STDP = t N Pairings Freq Normalized EPSC (%) Number of pairings (N) < t<0ms (post-pre pairings) 0< t<+30ms (pre-post pairings) Change in synaptic efficacy t (s) [Cui et al, 2015, J. Physiol.] [Graupner and Brunel, 2012, PNAS]

11 Different models of STDP STDP = t N Pairings Freq - Influence of the number of pairings in [Graupner and Brunel, 2012, PNAS] - Monotonic variation of the STDP curve as a function of the number of pairings Change in synaptic efficacy t (s) [Graupner and Brunel, 2012, PNAS]

12 Problematic and goals of the study [Cui et al, 2016, elife] : a biophysical model explaining this dependence but a complex model How to implementthis plasticity rule with a minimal model that explainsall the experimental data and that could be implementedin a network? Develop a simple phenomenological model taking into account the different mechanisms and dependence on the number of pairings Find parameters that would reproduce[cui et al, 2015, J. Physiol.] data Test the stability of this model

13 Different mechanisms for calciumbased STDP a b Pre CB1R Glu ecb LTP ecb LTD Post mglur ecb synthesis NMDA LTP CamKII AMPAr Ca NMDAr VSCC τ dρ ecb = ρ ecb (1 ρ ecb )(ρ ρ ecb ) dt +γ ecb LTP (1 ρ ecb ) Θ [Ca(t) θ ecb LTP ] γ ecb LTD ρ ecb Θ [Ca(t) θ ecb LTD ] +Noise ecb (t) τ dρ NMDA = ρ NMDA (1 ρ NMDA )(ρ ρ NMDA ) dt +γ NMDA LTP (1 ρ NMDA LTP ) Θ [Ca(t) θ NMDA LTP ] +Noise NMDA (t) Inspired from [Cui et al, 2016, elife]

14 Different mechanisms for calciumbased STDP a b Pre CB1R Glu ecb LTP ecb LTD Post mglur ecb synthesis NMDA LTP CamKII AMPAr Ca NMDAr VSCC τ dρ ecb = ρ ecb (1 ρ ecb )(ρ ρ ecb ) dt +γ ecb LTP (1 ρ ecb ) Θ [Ca(t) θ ecb LTP ] γ ecb LTD ρ ecb Θ [Ca(t) θ ecb LTD ] +Noise ecb (t) τ dρ NMDA = ρ NMDA (1 ρ NMDA )(ρ ρ NMDA ) dt +γ NMDA LTP (1 ρ NMDA LTP ) Θ [Ca(t) θ NMDA LTP ] +Noise NMDA (t) Inspired from [Cui et al, 2016, elife]

15 Different mechanisms for calciumbased STDP a Pre CB1R Glu ecb LTP ecb LTD Post mglur ecb synthesis NMDA LTP CamKII AMPAr Ca NMDAr VSCC Inspired from [Cui et al, 2016, elife] b τ dρ ecb = ρ ecb (1 ρ ecb )(ρ ρ ecb ) dt +γ ecb LTP (1 ρ ecb ) Θ [Ca(t) θ ecb LTP ] γ ecb LTD ρ ecb Θ [Ca(t) θ ecb LTD ] +Noise ecb (t) τ dρ NMDA = ρ NMDA (1 ρ NMDA )(ρ ρ NMDA ) 16 dt +γ NMDA LTP (1 ρ NMDA LTP ) Θ [Ca(t) θ NMDA LTP ] Calcium amplitude Noise NMDA (t)

16 Different mechanisms for calciumbased STDP a Pre CB1R Glu ecb LTP ecb LTD Post mglur ecb synthesis NMDA LTP CamKII AMPAr Ca NMDAr VSCC Inspired from [Cui et al, 2016, elife] b τ dρ ecb = ρ ecb (1 ρ ecb )(ρ ρ ecb ) dt +γ ecb LTP (1 ρ ecb ) Θ [Ca(t) θ ecb LTP ] γ ecb LTD ρ ecb Θ [Ca(t) θ ecb LTD ] +Noise ecb (t) τ dρ NMDA = ρ NMDA (1 ρ NMDA )(ρ ρ NMDA ) 16 dt +γ NMDA LTP (1 ρ NMDA LTP ) Θ [Ca(t) θ NMDA LTP ] Calcium amplitude Noise NMDA (t)

17 Different mechanisms for calciumbased STDP a Pre CB1R Glu ecb LTP ecb LTD Post mglur ecb synthesis NMDA LTP CamKII AMPAr Ca NMDAr VSCC Inspired from [Cui et al, 2016, elife] b τ dρ ecb = ρ ecb (1 ρ ecb )(ρ ρ ecb ) dt +γ ecb LTP (1 ρ ecb ) Θ [Ca(t) θ ecb LTP ] γ ecb LTD ρ ecb Θ [Ca(t) θ ecb LTD ] +Noise ecb (t) τ dρ NMDA = ρ NMDA (1 ρ NMDA )(ρ ρ NMDA ) 16 dt +γ NMDA LTP (1 ρ NMDA LTP ) Θ [Ca(t) θ NMDA LTP ] Calcium amplitude Noise NMDA (t)

18 Different mechanisms for calciumbased STDP a b Pre CB1R Glu ecb LTP ecb LTD Post mglur ecb synthesis NMDA LTP CamKII AMPAr Ca NMDAr VSCC τ dρ ecb = ρ ecb (1 ρ ecb )(ρ ρ ecb ) dt +γ ecb LTP (1 ρ ecb ) Θ [Ca(t) θ ecb LTP ] γ ecb LTD ρ ecb Θ [Ca(t) θ ecb LTD ] +Noise ecb (t) τ dρ NMDA = ρ NMDA (1 ρ NMDA )(ρ ρ NMDA ) dt +γ NMDA LTP (1 ρ NMDA LTP ) Θ [Ca(t) θ NMDA LTP ] +Noise NMDA (t) Inspired from [Cui et al, 2016, elife]

19 Different mechanisms for calciumbased STDP a b Pre CB1R Glu ecb LTP ecb LTD Post mglur ecb synthesis NMDA LTP CamKII AMPAr Ca NMDAr VSCC τ dρ ecb = ρ ecb (1 ρ ecb )(ρ ρ ecb ) dt +γ ecb LTP (1 ρ ecb ) Θ [Ca(t) θ ecb LTP ] γ ecb LTD ρ ecb Θ [Ca(t) θ ecb LTD ] +Noise ecb (t) τ dρ NMDA = ρ NMDA (1 ρ NMDA )(ρ ρ NMDA ) dt +γ NMDA LTP (1 ρ NMDA LTP ) Θ [Ca(t) θ NMDA LTP ] +Noise NMDA (t) Inspired from [Cui et al, 2016, elife]

20 Different mechanisms for calciumbased STDP a b Pre CB1R Glu ecb LTP ecb LTD Post mglur ecb synthesis NMDA LTP CamKII AMPAr Ca NMDAr VSCC τ dρ ecb = ρ ecb (1 ρ ecb )(ρ ρ ecb ) dt +γ ecb LTP (1 ρ ecb ) Θ [Ca(t) θ ecb LTP ] γ ecb LTD ρ ecb Θ [Ca(t) θ ecb LTD ] +Noise ecb (t) τ dρ NMDA = ρ NMDA (1 ρ NMDA )(ρ ρ NMDA ) dt +γ NMDA LTP (1 ρ NMDA LTP ) Θ [Ca(t) θ NMDA LTP ] +Noise NMDA (t) Inspired from [Cui et al, 2016, elife] Change in synaptic efficacy = S NMDA S ecb With S a sigmoid function

21 Dependence on the number of pairings + ecb LTP (1 ecb ) ecb LTD ecb d ecb dt = ecb (1 ecb )( ecb ) apple Z t Ca(t) ecb LTP Ca(t 0 ) dt 0 1 apple Z t Ca(t) ecb LTD Ca(t 0 ) dt 0 1 +Noise ecb (t) d NMDA = NMDA (1 NMDA )( NMDA ) dt apple Z t + NMDA LTP (1 NMDA LTP ) Ca(t) NMDA LTP Ca(t 0 ) dt 0 1 +Noise NMDA (t)

22 Dependence on the number of pairings

23 Theoretical solution using a mean-field approach - Mean-field approach as in [Graupner and Brunel, 2012, PNAS] for each independent mechanism - Resolution with Itoformula - ρ as a Gaussian distribution, computation of its mean and variance - Fit to the experimental data using Python scipy library

24 Application to anti-hebbian STDP a 0.1 Number of pairings t (s) Change in synaptic efficacy b c N Pairings = 14 N Pairings = 45 N Pairings = 100 d Change in synaptic efficacy t (s) t (s) t (s)

25 Comparison to experimental data STDP = t N Pairings Freq

26 How do the different mechanisms participate? a Only NMDA b Only endocannabinoids t (s) Change in synaptic efficacy t (s) Change in synaptic efficacy Number of pairings Number of pairings

27 Influence of frequency STDP = t N %34456' Freq 2.5 Change in synaptic efficacy t (s)

28 - A simple model with few parameters and 2 equations that reproduces experimental data of STDP - Predicitivemodel for frequency dependence [Feldman, 2012, Neuron]

29 - A simple model with few parameters and 2 equations that reproduces experimental data of STDP - Predicitivemodel for frequency dependence Future : Compare this model to other experimental situations in other parts of the brain Implement this STDP rule in networks

30 Acknowledgements Dynamic and Pathophysiology of Neuronal Networks Team at Collège de France: - Laurent Venance - Bertrand Degos - Marie Vandecasteele - Sylvie Perez - Iuliia Dembitskaia - Silvana Valtcheva - Alexandre Mendès - Willy Derrouseaux - Elodie Perrin The Mathematical Neuroscience Team at Collège de France: - Jonathan Touboul - Jérome Ribot - Tanguy Cabana - Richard Bailleul - Valentin Figué - Guillaume Hubert - Guillaume Penent And also : Michael Graupner Ilya Prokyn Hugues Berry