Memory maintenance in synapses with calcium-based plasticity in the presence of background activity

Abstract

Most models of learning and memory assume that memories are maintained in neuronal circuits by persistent synaptic modifications induced by specific patterns of pre- and postsynaptic activity. For this scenario to be viable, synaptic modifications must survive the ubiquitous ongoing activity present in neural circuits in vivo. In this paper, we investigate the time scales of memory maintenance in a calcium-based synaptic plasticity model that has been shown recently to be able to fit different experimental data-sets from hippocampal and neocortical preparations. We find that in the presence of background activity on the order of 1 Hz parameters that fit pyramidal layer 5 neocortical data lead to a very fast decay of synaptic efficacy, with time scales of minutes. We then identify two ways in which this memory time scale can be extended: (i) the extracellular calcium concentration in the experiments used to fit the model are larger than estimated concentrations in vivo. Lowering extracellular calcium concentration to in vivo levels leads to an increase in memory time scales of several orders of magnitude; (ii) adding a bistability mechanism so that each synapse has two stable states at sufficiently low background activity leads to a further boost in memory time scale, since memory decay is no longer described by an exponential decay from an initial state, but by an escape from a potential well. We argue that both features are expected to be present in synapses in vivo. These results are obtained first in a single synapse connecting two independent Poisson neurons, and then in simulations of a large network of excitatory and inhibitory integrate-and-fire neurons. Our results emphasise the need for studying plasticity at physiological extracellular calcium concentration, and highlight the role of synaptic bi- or multistability in the stability of learned synaptic structures.

Figure 1. Dynamics of the synaptic plasticity model with the in vitro and in vivo parameter sets.

(A) Pre- and postsynaptic spike trains generated as realisations of Poisson processes at 1/s. (B,C) The spike train in A induces large calcium transients (blue trace) with the in vitro parameter set ( and ; see Table 1). Whenever the calcium trace crosses the depression (cyan) or potentiation thresholds (orange), changes in the synaptic efficacy (green) are induced. (D,E) Same as in B,C but with calcium amplitudes corresponding to the in vivo case ( and ). The small calcium transients do not cross the depression/potentiation thresholds and no efficacy changes are observed. Note that the flat potential for is used here and that noise is set to zero for clarity, .

Figure 2. Memory decay for a single synapse with flat potential in the presence of uncorrelated pre- and postsynaptic Poisson firing.

(A,B) Temporal evolution of the mean synaptic efficacy in the presence of pre- and postsynaptic Poisson firing at 1/s for the in vitro (green in A) and the in vivo (light blue in B) parameter sets (mean shown for synapses). Blue and red lines show the mean dynamics as predicted by the Ornstein-Uhlenbeck theory. Grey lines show example traces of synaptic efficacy evolution in time. (C) Decay time constant as a function of the firing rate for in vitro and in vivo parameter sets. The blue and red lines show the calculated decay time constant, , from the OU theory. The points show exponential decay times obtained by fitting single exponential decay functions to the mean synaptic dynamics as shown in A and B illustrating that the OU theory correctly describes the full model behaviour. The cyan and orange dotted lines illustrate the derived power law behaviour, , between memory time scales and low firing rates (see text). The power reflects the number of spikes required to cross the plasticity thresholds, that is, for in vitro (cyan dotted line) and (orange dotted line) for in vivo case. (D) Asymptotic synaptic efficacy as a function of the firing rate for in vitro and in vivo parameter sets. The lines show the calculated asymptotic value, , from the truncated OU theory () for in vitro (blue line) and in vivo (red line) cases. Note that at high frequencies saturates at a value equal to , since both depression and potentiation terms are active in the high calcium region. The points show steady-state values obtained by fitting single exponential decay functions to the mean synaptic dynamics as shown in A and B (green: in vitro; light blue: in vivo).

Figure 3. Memory decay for a bistable synapse in the presence of uncorrelated pre- and postsynaptic Poisson firing.

(A) Steady-states of synaptic efficacy as a function of firing rate for the in vitro (blue) and the in vivo (red) parameter sets. Stable states are shown by solid lines and unstable states by dotted lines. Synaptic efficacy is bistable at low rates (/s for in vitro and /s for in vivo) and monostable at high firing rats. The effective potential of synaptic efficacy is shown for three firing rates (0.1/s - magenta line; 1/s - orange line; 2/s - green line) and the in vivo parameter set in the inset (firing rates indicated by vertical lines). (B) Decay time constant as a function of the firing rate for the in vitro and the in vivo parameter sets. For the in vivo parameter set below /s, the bistability greatly extends memory time scale compared to a synapse with flat potential (red line) and can be predicted using Kramers escape rate (magenta line). The vertical dashed line illustrates the frequency at the in vivo bifurcation point. For the in vitro parameter set, the bistability has no influence on decay time constants for firing rates above 0.1/s. The points show exponential decay times obtained by fitting single exponential decay functions to the mean synaptic dynamics. (C) Individual synaptic efficacy traces for the in vivo parameter set at 1/s pre- and postsynaptic firing. The synapses remain in the upper potential well for a long time and stochastically cross the potential barrier to the low efficacy state. (D) Averaged synaptic efficacy trace of many synapses for the in-vivo parameter set at 1/s. The bistability extends the memory time scale from hours for a flat potential to days.

Figure 4. Steady-state behaviour of a recurrent network with plastic synapses between excitatory neurons.

(A) Firing rate mean-field predictions compared with network simulation results. The mean-field theory predicted firing rate is higher (black line) than the actual firing rate of the excitatory neurons (green dots) in the recurrent network of 8000 exc. and 2000 inh. LIF neurons. Network simulation with fixed synapses yield a good match with the theory (blue dots). (B) Average synaptic weight prediction compared with asymptotic average synaptic weights in the network simulation. The observed average synaptic efficacy of excitatory to excitatory connections is smaller (mustard dots) than the theoretical prediction (black line). Even when using the asymptotic firing rate of the network in the calculations (green dots), the average synaptic efficacy is overestimated by the theory. (C) Mean and standard deviation of synaptic weights vs. firing rate for independent LIF neurons (magenta), networked LIF neurons (green) and LIF neurons in a network in which actual weights are held constant but we examine how their efficacy would have evolved in the presence of observed firing (blue dots). Asymptotic synaptic weights for LIF neurons underestimate the efficacy predicted by the theory (blue line). (D) Average synaptic weight vs. firing rate for independent LIFs with different reset potentials. The analytical prediction of the asymptotic synaptic weight based on Poisson firing is shown by the blue line (same as in C). The reset potential in the LIF model, , has a marked influence on the observed average synaptic efficacy. Depolarised/hyperpolarised reset potentials (e.g. −55/−70 mV, cyan/green dots) lead to an over/under-representation of short ISIs (left/right inset) compared to Poisson neurons (red line in insets). ISI histograms in inset are shown for LIF neurons firing at 1/s.

Figure 5. Memory decay for a subset of potentiated synapses in a recurrent network with the in vitro parameter set.

(A) Temporal evolution of the average excitatory (red) and inhibitory (blue) firing rate. A network of 10,000 LIF neurons is initialised at the theoretically predicted steady-state and simulated for 20 min real time. (B) Temporal dynamics of synaptic efficacies in the network. The majority of synapses are initialised to the theoretically predicted asymptotic synaptic efficacy (mean: magenta; single synapse example: dark gray). A randomly selected subset of 5% are set to 1 at the beginning of the simulation (mean: green; single synapse example: light gray). (C) The exponential decay time constant of the potentiated synapses. The value obtained from fitting a single exponential to the mean decay (green dots) is well approximated by the analytically calculated decay time constant from the OU process (Eq. (3)). Introduction of a double-well potential does not modify the memory time constant for the in vitro parameter set (orange stars). The slight deviation of the decay time constants with respect to the OU theory, that is, the network decay time constants are slower, are due to the LIF firing statistics as can be seen from the comparison with independent LIF neurons (magenta dots).

Figure 6. Memory decay for a subset of potentiated synapses in a recurrent network with the in vivo parameter set and double-well potential.

(A) Temporal evolution of the average excitatory (red) and inhibitory (blue) firing rate. A network of 10,000 LIF neurons is initialised at the theoretically predicted steady-state and simulated for 120 min real time. (B) Temporal dynamics of synaptic efficacies in the network. The average dynamics of the 95% initialised in the DOWN state (blue) and the 5% initialised in the UP state (red) is shown. The shaded gray region represents the range of values visited by synapses in the UP and in the DOWN state populations, indicating that no transition occurs.

Figure 7. Possible potentiation and depression threshold crossing cases of the calcium trace (blue lines) between events at time and .

The six possible cases are depicted with respect to the location of the potentiation, (orange dashed line), and the depression thresholds, (cyan dashed line).

Figure 8. Example of network firing in asynchronous irregular state.

A sample of 1000 neurons from the network shows irregular spiking behaviour in the raster (top) and the averaged firing rate of all 8000 excitatory neurons is steady artabound 1/sec (bottom).