A Calcium dependent model of synaptic plasticity (CaDp)

Presynaptic rate induced: Various different protocols for inducing bidirectional synaptic plasticity Pairing induced (postsynaptic voltage clamp) Spike time dependent plasticity (STDP) Feldman, 2000 Markram et. al. 1997

Can a single model, based on a limited set of assumptions, account for the various induction protocols? Approach: Find a minimal set of assumptions that can qualitatively account for the various forms of induction.

Assumption 1: The calcium control hypothesis. The idea that calcium levels control the sign and magnitude of synaptic plasticity has been around for a while (Lisman, 1989; Bear et. al., 1987; Artola et. al. 1990) ΔWΔWΔWΔW LTD Ca LTPθdθd θpθp I. A Unified theory of NMDA Receptor-Dependent synaptic plasticity Ω function

Whereand *This equation can be derived from a lower level biophysical formulation. (Castellani et. al. 2001, Shouval et. al. 2002) The calcium control hypothesis, is a generalization of this equation Ca (  M)  d  p  Ca (  M)  sec the rate function is: has the form *

Assumption 2: NMDA receptors are the primary source of calcium influx to spines during synaptic plasticity ( Sabatini et. al 2002 ). Voltage dependence of NMDAR (Jahr and Stevens, 1990) Standard assumptions Fraction of open NMDAR I Ca

Ligand binding kinetics – sum of two exponentials with different time constants (Carmignoto and Vicini, 1992) Calcium Dynamics- first order ODE NR2A+NR2B In these examples NMDA receptor kinetics- sum of two exponents

Pairing Induced Plasticity Voltage clamping postsynaptic neuron while stimulating presynapticaly at 1 Hz. Examples LTP/LTD curve W W

Bi and Poo, 1998 Spike time dependent plasticity (STDP) STDP Curve

For the calcium control hypothesis to account for STDP it is necessary that: For (post-pre) the calcium influx is higher than at baseline ( ) For ( pre-post) the calcium influx is higher than at ( )

Axon: output Action potentials | || | || | | | | | | || | | Neuron – cell body Dendrite: input Synapse Back propagating action potentials

Assume a narrow spike (Width 3ms) Problems: No difference between baseline and post-pre Only a small elevation in Ca for pre-post Back spike – assume width 3ms

Assumption 3: The Back Spike has a slow component (long tail). narrow spike (3ms) spike with long tail (width 25 ms) An example of a BPAP recorded by C. Colbert from a hippocampal dendrite (slice, from 180 gm Sprague Dawley rat at 31 o C, 150µM from soma)

Back Spike with long tail (tail width 25ms) Problems solved Ca level in post-pre larger than at baseline. Larger elevation of Ca in pre-post condition.

BPAP with wide tail (ms) Similar results: Karmarkar and Bunomano, 2002; Abarbanel et. al. 2003; Kitijima and Hara, 2000

Nishiyama et. al. Nature, 2000Bi and Poo J. Neurosci Wittenberg and Wang, 2006 Froemke and Dan, 2005 Does the second LTD Window exist?

Frequency dependence of the STDP curve This occurs due to temporal integration of the calcium transients, and is not simply a consequence of the low frequency form of STDP. Significance: The (low frequency) STDP curve can not serve as a basis for STDP in general, let alone other forms of synaptic plasticity. Stability of STDP depends on the small excess of LTD over LTP, this occurs only at low frequency.

STDP in Neocortex Thus, with different parameters our model can also account for neocortical STDP.

NMDA receptor dependent Calcium influx NMDA receptor kinetics NMDAR kinetics and subunit composition are plastic. Subunit composition, current duration and magnitude change during development and are activity dependent (Carmignoto and Vicini, 1992; Quinlan et. al. 1999, Philpot et. al. 2001) NR2A+NR2B

Dependence of plasticity on NMDA receptor kinetics I f is the magnitude of the fast component of the NMDA current. Large I f fast current, and small integrated calcium influx Pairing inducedPresynaptic rate induced I f =0.25 (green)I f =0.5 (blue)I f =0.75 (red) We assume that the peak NMDAR current is constant ( I f + I s =1)

Dependence of plasticity on NMDA receptor kinetics I f =0.25 (green)I f =0.5 (blue)I f =0.75 (red) I f is the magnitude of the fast component of the NMDA current. Large I f fast current, and small integrated calcium influx We assume that the peak NMDAR current is constant ( I f + I s =1) STDP

Width of BPAP tail- correlates with shape of STDP

Froemke and Dan preliminary data (Soc. Neurosci. 2002)

I. CaDP- A simple theory that accounts for the induction of Calcium-dependent bidirectional synaptic plasticity. II. Stochastic calcium transients affect the induction of plasticity with CaDP. III. Receptive field formation with CaDP and metaplasticity.

Summary I: A simple theory, with a small number of assumptions can account for the various induction paradigms Major Assumptions: I.Calcium control hypothesis II.Calcium influx through NMDAR III.Slow tail to the BPAP Novel Consequences: I.The width of LTD and magnitude of LTP correlate with the width of the BPAP tail. II.A novel pre-post form of LTD. III.Frequency dependence of STDP.

II. Stochastic Calcium transients effect the shape of plasticity curves. Up to now we have simulated numerically the calcium transients, disregarding the the effect of stochastic fluctuations. I. Analytical derivations: (a) A closed form solution of the mean calcium transients. (b) The variability of calcium transients due to a limited number of postsynaptic NMDA receptors II. Simulations of STDP plasticity curves with stochastic fluctuations of calcium transients. What happens to pre-post LTD?

Analytic calculation of the mean of calcium transients: Simplifying assumptions: (a) NMDAR dynamics: Single Exponential Component (b) BPAP dynamics: Single Exponetial Component (c) Linear approximation to H(v ) H(v) Linear approximation V (mV) time (ms)  N Prob open H(v) One component t (ms) mV BPAP Two components

Calcium influx: Formal Solution: According to our previous assumptions I ca (t) becomes a sum of exponentials (Yeung et. al 2004), and the integral can be solved.

Consequences of two component back spike Effect of the slow component of the back spike: Effect of Calcium time constant: I s (no units) Peak [Ca +2 ] (  M) Ca pre|post Ca post|pre Ca pre  Ca (ms) Relative Peak [Ca +2 ] (Ca post|pre -Ca pre )/Ca post|pre (Ca pre|post -Ca post|pre )/Ca pre|post Slow component separates Ca pre from associative calcium signals A fast calcium time constant enhances the difference between post|pre and pre|post

Calculation of average calcium transients BPAP composed of one or two exponential components Results two components Time (ms) [Ca +2 ] (  M) Δ t=10 Δ t=-10 Pre only  t (ms) I peak +/- (  M / ms)  N magnitude of slow component (ADP) magnitude of fast component τfτf τsτs Two components One component τBτB

 t (ms) I peak +/- (  M / ms)  B  N LTD LTP LTD Consequences for synaptic plasticity LTD LTP

1.Release of Glutamate is a stochastic process. 2. Opening and closing of NMDAR is a stochastic process Analysis of the variability of calcium dynamics, taking into account the stochastic properties of synaptic transmission:

Stochastic properties of NMDA receptors: u o k1Gk1G k -1 A very simple 2 state Markov model (= 1 NMDAR time constant) u – unbound o – open + bound G- Glutamate τ N =1/k -1 Other variables: M, number of NMDAR μ, average fraction of NMDAR that open as a result of one presynaptic AP The open probability has the form: Therefore the mean transients are:

Results: (assume number of NMDA receptors, is M=10) t (ms) [Ca] (  M)  t =10 ms mean(Ca) std(Ca) t (ms) std(Ca) mean(Ca)  t =-10 ms t (ms)  t =60 ms mean(Ca) std(Ca) CV=std/mean, at peak of Ca transient Number of NMDAR

Results: For M= 10, τ =75 ms CV(60)/CV(-10)=1.5 Experimentally the number of NMDAR in spine is not well known. However there are estimates of ~10 (Racca et. al. 2000) Qualitatively confirmed in simulation using a Markov model of NMDA receptors Closed Open Bound 1 Bound 2 Desensitized 1 Markov model as in Lester and Jahr, (1992), Franks et. al. (2003).

Calcium transients with and without ‘noise’ Presynaptic failuresPostsynaptic variability of magnitude

Stochastic implementation CV – given there is release Distribution of Ca transients

Effect of noise on STDP curves Shouval and Klantzis, J. Neurophys, 2005 a b W M=10  =15 ms  =57 ms 1 M=  t (ms) W M=  t (ms) M=3200

Why? Ω Ca Δt=60 ms t (ms) Ca (  M) mean(Ca) std(Ca) Δt=-10 ms Ca (  M) std(Ca) mean(Ca) Ω Ca Ω Ca (  M) mean(Ca) std(Ca) Δt=10 ms

II. Summary, stochastic analysis: The NMDAR time constant controls the width of pre-post LTP, and the BPAP time constant controls post-pre LTD T he relative variability depends on the number of NMDAR at each spine, such that: CV M Variability of calcium transients, for M small enough, reduces the magnitude of pre-post LTD more than of post-pre LTD.