Maintenance of memory in a cortical network model 

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Synaptic plasticity experiments: hippocampus and neocortex

Initial experiments demonstrating synaptic plasticity were performed by Bliss and Lømo (1973), in which they demonstrated, for the first time, the ability to change the excitatory postsynaptic potential (EPSP) in a population, by extracellular stimula-tion of a bundle of presynaptic fibres. The effect was demonstrated in the hippocam-pus of an anaesthetised rabbit and in a subsequent paper for an unanaesthetised animal (Bliss and Gardner-Medwin 1973). They also demonstrated an increase in the excitability of the postsynaptic population, an effect which is related to but distinct from synaptic potentiation (Daoudal and Debanne 2003). They called the increase in synaptic transmission potentiation. The first demonstration of the oppo-site effect to potentiation, called depression, was by Lynch et al. (1977) where they demonstrated in hippocampal slice that the potentiation of one set of synapses led to the depression of other synapses terminating on the same set of postsynaptic cells.
Figure 1.4. Bliss and Lømo (1973). Early experimental demonstration of synaptic potentiation. The test pathway demonstrated a potentiation which lasted for hours (black circles) whereas the control pathway demonstrated no change (open circles).
Dudek and Bear (1992) demonstrated that presynaptic activity which does not provoke a postsynaptic spike can lead to synaptic depression. This work illustrated a clear frequency dependence of the effect and also a reversability via what had previously been seen purely as potentiation protocols, suggesting that potentiation and depression are opposing rather than independent effects. In a follow-up paper, Dudek and Bear (1993) confirmed that LTP is the reversal of LTD and vice-versa.
Figure 1.5. Debanne et al. (1994). The first demonstration of a relative timing dependence between pre- and postsynaptic processes in order to produce potentiation/depression. (A) In a paired pre- post- protocol we observe LTP. (B) However, in an unpaired input pathway, which was stimulated 800ms following the paired input pathway to the same cell, LTD was observed. (C) The LTD was not dependent on a prior stimulation of a the paired pathway but only on the prior depolarisation of the postsynaptic cell.
Figure 1.6. Bi and Poo (1998). The spike pair based spike timing dependence plasticity timing window. A pre- before a post- spike leads to potentiation. Whereas post- before pre- leads to depression.
In the 1990s there quickly followed a succession of papers, moving from previously exclusively extracellular techniques to intracellular recording, first, and subsequently also stimulation, demonstrating a precise timing dependence to synaptic plasticity.
An early paper (Debanne et al. 1994) demonstrated that pairing a postsynaptic depolarisation with a subsequent presynaptic stimulus leads to LTD (see Fig. 1.5), a result which had long been predicted as a theoretical corollary of Hebb’s potentiation rule (Hebb 1949). In the same paper they also confirmed the dependence of LTD on postsynaptic NMDAR activation, suggesting a dependence on postsynaptic calcium influx via voltage-gated calcium channels. The same group followed-up, (Debanne et al. 1996) with further confirmation that, in fact, for single postsynaptic action potentials the precise timing of presynaptic relative to postsynaptic action potentials is extremely important for synaptic plasticity, whereas in the case of bursts the frequencies of spiking are more important. Using dual whole-cell recordings (previous work used extracellular presynaptic stimulation) Markram et al. (1997) demonstrated that presynaptic stimulation alone, which did not produce a postsynaptic action potential, does not lead to changes in synaptic efficacy. Instead there is a requirement for both pre- and postsynaptic activity in order to induce synaptic changes. This work implies a requirement for a backpropagating action potential in the postsynaptic cell for such a mechanism to work. They also further confirmed the pre- before post-paradigm, implied by Hebbian theory, leading to LTP and post- before pre- leading to LTD. Finally, Bi and Poo (1998) in hippocampal cell cultures thoroughly investigated the relative pre- post- timing dependence of synaptic plasticity demonstrating that synaptic changes can be induced by repeated single activation, at a low frequency, of both pre- and postsynaptic action potentials with only the relative timing between the potentials dictating the sign and magnitude of the changes (see Fig. 1.6). Closely separated action potentials were shown to lead to larger changes, while sufficiently largely separated action potentials lead to no synaptic change. Following the Hebb paradigm, pre- before post- synaptic potentials were shown to lead to LTP whereas post- before presynaptic potentials led to LTD.
Markram et al. (1997) reported that at protocol repetition frequencies lower than approx. 10Hz zero synaptic change occurred. Exploring this Sjöström et al. (2001) found that in fact, while a typically LTP pairing protocol leads to zero change at these frequencies, an LTD protocol still leads to depression even at low frequen-cies. Through a series of experiments they further demonstrated that this frequency dependence, on LTP, can be overcome by generating a larger postsynaptic EPSP, either through activation of more synaptic inputs or via a stronger stimulation, or alternatively via experimental postsynaptic depolarisation. In (Lisman and Spruston 2005) it was postulated that this is likely due to the requirement for a ‘build-up’ in synaptic current charge, due to repetition, in order to fully release the postsynaptic magnesium block, on the NMDA receptors, allowing for a subsequent opening of the voltage activated calcium channels. They reasoned that the heretofore accepted influence of the backpropagating action potential may not in fact be an enabler of synaptic plasticity due to its attenuation at higher frequencies. Citing Golding et al. (2002) as evidence, they argue that it is a build-up of postsynaptic dendritic depo-larisation which is a necessary signal for synaptic plasticity, rather than an actual postsynaptic action potential.
A number of groups have looked at the effects of higher numbers of pre- and post-synaptic spikes, than that originally examined in the basic STDP experiments, on synaptic plasticity. In (Froemke and Dan 2002) the authors explore a paradigm of a single presynaptic spike sandwiched closely in time between two postsynaptic spikes as well as the inverse setup; two presynaptic spikes surrounding a single postsynaptic spike. They find that the net synaptic plasticity cannot be explained as any kind of linear sum of the independent LTP and LTD effects, which can be expected by a single pre- spike followed by a post- spike and vice-versa. They propose a model, based on their observation that the time difference between the first pre- spike and the first (or only) post- spike is the greatest predictor of synaptic plasticity out-come, which requires that a presynaptic spike suppresses the influence of subsequent presynaptic spikes occurring within a short time window. In (Froemke et al. 2006) they extended this work, examining the effects of a burst of length 5 in combination with an isolated pre- or postsynaptic spike, and subsequently with a burst of equal length. By modifying their original model somewhat they find that suppression of the influence of non-initial spikes in a burst on the plasticity rule appears to explain the data. Taking a similar approach, this time using cultured hippocampal neurons, Wang et al. (2005) showed a definite asymmetry in the LTP and LTD learning rules. Using a triplet protocol they demonstrated that a typically potentiating spike pair followed by a typically depressing paring (in this case the postsynaptic spike is shared between both pairings) leads to to a cancellation of the two processes, however re-versing the protocol order leads to potentiation; suggesting that LTD can cancel LTP but LTP can completely overrule LTD. They then proceed to demonstrate the same effect for a quadruplet of sufficiently closely occurring spikes. Wittenberg and Wang (2006) further explore higher dimensional effects on the CA1 plasticity rule, at low extracellular calcium concentrations. At low frequency stimulation they demonstrate a complete absence of LTP effects, whereas LTD is produced for any closely paired pre- and postsynaptic spikes. Repeating this experiment at 5Hz leads to some cells potentiating whereas others depress for similar relative spike timings, however the addition of a second postsynaptic spike, along with the 5Hz tetanic frequency finally restores LTP for anti-causally oriented spike pairs. Finally, they demonstrate that LTP has a lower repetition dependence than LTD in order to produce effects of a similar magnitude.
Indications that synaptic plasticity saturates were first evident in (Bliss and Lømo 1973) where, furthermore, we see clear evidence that LTP may be an all-or-nothing process. Many subsequent experiments (cf. Debanne et al. (1994); Sjöström et al. (2001)) demonstrated that the two reversible processes, LTP and LTD, appear to have two distinct stable states, at least for the one hour duration following a plasticity induction experiment. In (Petersen et al. 1998) they demonstrated that potentiation of CA3 to CA1 synapses appears to be an all-or-none process, leading to saturation once the binary step-up in synaptic efficacy has occurred. As part of this demonstration it is important to note that different individual synapses have different individual efficacies and different thresholds to potentiation. This means that in a cooperative, distributed, synaptic system we will see synaptic behaviour which often resembles an analogue distribution of synaptic efficacies. The all-or-none nature of bidirectional plasticity was further explored in a pair of papers (O’Connor et al. 2005a; O’Connor et al. 2005b). In the Journal of Neurophysiology paper they examine the kinase and phosphatase pathways required for LTP and LTD. In the PNAS paper they clearly demonstrate that plasticity events are discrete, with only two states for a given synapse. Furthermore, the individual synaptic states are heterogenous again demonstrating a potential method of graded memory storage via different synapses.
The detailed biomechanical machinery responsible for synaptic plasticity involves a dizzying array of chemicals and pathways. An early substance which was found to be vital for synaptic plasticity was calcium. Malenka et al. (1988) demonstrated, via a photo-induced increase in postsynaptic calcium concentration, that elevated calcium is a necessary and sufficient condition for postsynaptic LTP. This result had already been indicated in (Lynch et al. 1983) where intracellular EGTA was shown to block LTP. In practice, LTP activation is believed to be via the postsynaptic NMDA receptors, as blocking their activity via APV prevents LTP (Collingridge et al. 1983). In fact, NMDA receptors are believed to be critically implicated in LTP, requiring the coincidence of a presynaptic spike, inducing glutatmate release, and postsynaptic depolarisation, releasing the magnesium block in the postsynaptic NMDA receptor, for induction (Bliss and Collingridge 1993). Calcium dependence of LTD was sub-sequently demonstrated by Neveu and Zucker (1996), again via photolysis of caged calcium compounds. In a follow-up paper, Yang et al. (1999) demonstrated that while LTP was sensitive to brief but high increases in calcium, LTD is induced via longer lasting intermediate levels of calcium increase. From the timing of synap-tic plasticity experiments, the fact that elevated calcium concentrations decay much faster than the ultimate induction processes, it is clear that the ultimate actors of synaptic plasticity are downstream of any calcium activation and probably involve either, calcium dependent proteases, protein kinase C (PKC) or calcium-calmodulin dependent protein kinase II (CaMKII).

Synaptic plasticity models: hippocampus and neocortex

The original model of synaptic plasticity appears, somewhat informally, in Hebb’s seminal work (Hebb 1949). It expresses a form of coincidence detection, whereby synapses joining neurons which ‘fire together’ either come into existence or become stronger; the details are left for the reader to imagine. There is no explicit definition of what happens to existing synapses joining neurons which are not firing together. However, over time it has come to be assumed that synapses which connect neurons which are not ‘firing together’ probably need to decay in strength and ultimately to disappear. This is in order to avoid a system where synaptic efficacies only grow, leading to increased firing rates and further synaptic strength increases.
In (Sejnowski 1977) the problem of interpreting Hebb’s rule was approached us-ing a covariance model. In this it was proposed that the influence of presynaptic activity on postsynaptic membrane potential is governed by the way in which the two membrane potentials or spiking processes covary. Kohonen (1982) proposed a firing rate based approach which saturates in order to avoid unrealistic growth of the synaptic efficacy. A power series expansion of this model was taken by Oja (1982), giving a learning rule which learns due to Hebbian type coincident activity but which has a normalising leak term. This model was then shown to implement a principal component analyser when embedded in a network. An alternative rate based plas-ticity rule was presented in (Bienenstock et al. 1982) where they showed that the introduction of a sliding threshold to such a rule allows them to account for such developmental traits as orientation tuning curves and ocular dominance in the visual cortex (see Fig. 1.8).
The Hopfield (1982) model of neuronal networks combines binary threshold neu-ronal units with a network structure which resembles spin-glass models studied in statistical physics, where synapses are updated in response to input patterns, rep-resenting memories to be recalled. This model proved tractable to mathematical analysis leading to a calculation for memory capacity in (Amit et al. 1985). They showed that, for a low synaptic noise, memories can be stored with perfect recall un-til the memories represent more than a critical proportion of the size of the network, after which a catastrophic failure of the memory process may occur. Nadal et al. (1986) showed that by modifying the Hopfield (1982) learning rule new memories can continue to be stored, indefinitely, at the cost of erasure of older memories. Finally, Tsodyks and Feigel’man (1988) proposed a modified learning rule for the Hopfield (1982) network, which featured greatly enhanced storage capacity for correlated pat-terns, when activity levels are low.
Rate based model development was driven by the experimental state-of-the-art at the time. Experimental results typically involved external stimulation of bundles of axonal fibres and recordings of population dynamics (cf. Bliss and Lømo (1973)). With the development of reliable intracellular recording techniques, the 1990s saw a switch to precise spike timing driven experiments and a commensurate switch in the modelling community. A forerunner however was Gerstner et al. (1993) where they showed that a Hebbian rate-based model, especially in a system with a rea-sonable variance in transmission delays, was not able to capture temporally coded information, whereas a spike-timing based rule can do so to a very high precision. This was followed up in (Gerstner et al. 1996) where the model was applied to ex-plain the ability of barn owls to localise sound with a very high temporal precision despite apparently longer time constants in their auditory system than the latency between the two input channels. In (Song et al. 2000) they demonstrated that a simple spike-timing dependent plasticity (STDP) rule in a recurrent network nat-urally leads to selective weight potentiation. Synapses which are most involved in postsynaptic spiking tend to get potentiated whereas other synapses slowly get de-pressed. This demonstrates not only a clear motivation for the existence of STDP in synapses but a potential method of homeostasis of incoming weight to a single neuron, without the requirement for a global signal. In order to achieve a weight dis-tribution more closely matching that seen in experiments Van Rossum et al. (2000) introduced a weight dependence on the potentiation term, meaning that stronger weights get potentiated less than weaker weights. This leads to a unimodal distribu-tion but requires an extra ‘leak’ term in order to maintain a target postsynaptic firing rate. The same authors analysed this model further in (Van Rossum and Turrigiano 2001) with the slight alteration, based on experimental observations, where there is no longer a weight dependence on depression, they demonstrate that such a rule leads to a stable distribution of synaptic weights under random background firing and that it will selectively learn for correlated inputs. Examples of the distributions obtained from additive and multiplicative STDP rules can be seen in Fig. 1.9, taken from Billings and Van Rossum (2009), a paper which will have relevance to our work in Chapter 3 as it pertains to memory maintenance and decay in these two models of STDP.
As experiments developed allowing the investigation of precise spike-timing ef-fects on synaptic plasticity, it also became possible to more accurately explore the previously described phenomenon of short-term synaptic plasticity (Zucker 1989). This is a, typically, presynaptic effect, whereby there is a delay in replenishing presy-naptic vesicles thus leading to a decrease in the observed effectiveness of the synapse in inducing an excitatory EPSP. The typical model of this process is the depletion-renewal model of Tsodyks and Markram (1997). Short-term facilitation was added to this model in (Tsodyks et al. 1998). Short-term plasticity was developed into a theory of a synaptic mechanism/trace of working memory storage in (Mongillo et al. 2008). In this thesis, we will focus only on models and mechanisms of long-term plas-ticity, their short-term counterparts are mentioned here for completeness and also to illustrate a point at which the literature separated into short-term and long-term components.
Synaptic memory processes involve discrete processes, such as an increase in vesi-cle density presynaptically or in AMPA receptors postsynaptically (Lisman 1985). This has led to a number of theories of synaptic plasticity which involve discrete synaptic weights, or at least the use of discrete stable states connected by a con-tinuous potential well. In (Amit and Fusi 1994) learning is modelled by a Markov process, representing the probability of transition between stable states, giving rise to a palimpsest memory process capable of continually learning new memories while decay of old memories is dependent on subsequent patterns presented. Fusi et al. (2005) advanced the concept of bistability in synapses to involve multiple internal (meta-)states in a synapse, which may refer to amenability towards plasticity in either direction, with two or more discrete states visible to the network. Versions of this model have been found to have properties which contribute to both high memory capacity in a network and long memory time scales (Roxin and Fusi 2013). While not strictly discrete, the strong bistability in the Tag-Trigger-Consolidate model of Clopath et al. (2008) can be interpreted as a binary synapse and the underlying mechanism demonstrates an attractive link between the abstract model of Fusi et al. (2005) and the underlying biology involving early- and late-phase plasticity driven by protein processes.
When typical synaptic plasticity models are implemented naively they may give rise to pathological behaviour. That is, an increase in synaptic strength will lead to increased postsynaptic spiking/excitability which tends in turn to lead to fur-ther increases in synaptic strength. A number of solutions to this problem have been proposed and fall generally under the term homeostatic plasticity or regulation (Turrigiano and Nelson 2000; Turrigiano and Nelson 2004). In brief, this means that there is some kind of normalisation process, which operates across all, or a subset, of the synapses inputting to an individual postsynaptic neuron, generally at a much slower time scale than that of spike-timing dependent plasticity, which leads to a moderation in postsynaptic spiking activity. Such an idea is implicitly present in the sliding threshold, at the individual synapse level, for the BCM rule (Bienenstock et al. 1982). Toyoizumi et al. (2005) developed a synaptic plasticity rule based on the principle of maximisation of information between spiking neurons, a process which requires that homeostasis maintains the postsynaptic firing rate as close to its mean as possible, finding that such a rule has the same properties as the BCM rule in the absence of refractoriness. The weight dependent potentiation in the Van Rossum et al. (2000) model can be seen as an early attempt to incorporate such a process into a spike-timing plasticity model. This model does not necessarily prevent run-away excitation but it does at least greatly reduce the speed of its activation allowing for network processes to intervene. The Tag-Trigger-Consolidate model of Clopath et al. (2008) contains a more explicit implementation of homeostatic synaptic plasticity via a process called cross-tagging, a process global to the postsynaptic neuron although it could also be implemented locally to a dendritic branch.
Figure 1.10. Shouval et al. (2002). A calcium-based plasticity rule is capable of reproducing both the postsynaptic depolarisation dependence (A) and frequency dependence (B) of STDP. This particular rule predicts LTD for both positive and negative relative spike timings and LTP only for positive offsets (C).
The initial spike-timing dependent plasticity models were purely phenomeno-logical models, eventually there was a move to base these models more closely on underlying physiological processes. As calcium had been shown to be necessary and sufficient for LTP (Malenka et al. 1988) and LTD (Neveu and Zucker 1996) it was natural for one of the first such models to use calcium as a central element (Shou-val et al. 2002). In their model (see Fig. 1.10), Shouval et al. implement calcium influx postsynaptically via NMDA receptors, they do this by assuming that back-propagating action potentials induce long lasting after-depolarisations which release the magnesium blocks which had been preventing calcium entry via the NMDA re-ceptors. This model can, at least qualitatively, fit the postsynaptic depolarisation dependence (Cummings et al. 1996), the frequency dependence (Bliss and Lømo 1973; Dudek and Bear 1992) and precise spike-timing based experiments (Bi and Poo 1998; Markram et al. 1997) in the early literature. It does however predict a late period of LTD plasticity, where a postsynaptic spike follows a presynaptic spike by approx. 100ms, over which there is much debate as to the real existence. The advantage of such a model is that it allows us to make precise predictions about experimentally verifiable variables in order to understand if our suspicions about the processes underlying synaptic plasticity are valid. A more detailed calcium depen-dent model was presented in (Rubin et al. 2005), in which they explicitly attempt to model the pathways regulating CaMKII in a single spine using a series of dif-ferential equations. Two of the equations filter the calcium concentration leading to LTP and LTD of the synapse at different calcium concentrations, with a strong duration dependence. They propose that a system of calcium detectors, which in-clude a duration dependence for LTD, must be combined with a veto process on depression in order to be sufficiently robust to fit all experimental data. Such a veto might be compared with the competition of kinases and phosphatases in the CaMKII phosphorylation-dephosphorylation cascades.
In recent times a number of models have been devised which attempt to go be-yond basic pair-wise spike-timing dependent plasticity, to capture higher dimensional features in spike patterns. An early such model was presented in (Sjöström et al. 2001) where they incorporated firing rate, depolarisation and relative spike timing into the synaptic plasticity rule. Fitting the data presented in the same paper they found their best model was one in which a spike which participates in an LTP pro-cess should not contribute to a separately calculated LTD process. An alternative approach was presented in (Froemke and Dan 2002) and more thoroughly developed in (Froemke et al. 2006) and was based upon experimental results which appeared in the same papers. In this model the underlying idea is that subsequent spikes in a burst have a lesser effect on the overall plasticity change. A model very much in keeping with the original idea of pair-wise plasticity was the model of Pfister and Gerstner (2006) which moves to incorporating the three most recent spikes and their relative timings in order to predict plasticity changes. The emergence of this model was clearly influenced by the emergence of triplet based experiments such as Wang et al. (2005). An attractive feature of this model is that it can be directly mapped to a BCM type rule if spiking is assumed to be a Poisson process pre- and postsynapti-cally (Pfister and Gerstner 2006). A more biologically inspired model was presented by Clopath et al. (2010), in which thresholded low-pass filtered and instantaneous traces of postsynaptic membrane voltage are multiplied by learning constants upon presynaptic spiking. This brings the postsynaptic voltage dependence of synaptic plasticity (Sjöström et al. 2001) to the fore in the model and combines it with traces which can variously be explained as NMDAR activation, endocannabinoid release and postsynaptic calcium concentration. The model we will concentrate mostly upon, in this work, is the calcium-based plasticity rule of Graupner and Brunel (2012), which attempts to combine the direct biological relevance of a calcium-based rule with cer-tain features, such as thresholding of the calcium trace, which capture higher order features of spike trains. This rule is based entirely upon duration above the relevant calcium thresholds in order to dictate plasticity outcomes.

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Table of contents :

1 Introduction 
1.1 Synaptic plasticity experiments: hippocampus and neocortex
1.2 Synaptic plasticity models: hippocampus and neocortex
1.3 Synaptic plasticity experiments: cerebellum
1.4 Summary
2 Parallel-fibre to purkinje cell plasticity 
2.1 Model Description
2.2 Methods
2.2.1 Synaptic plasticity model
2.2.2 Simulating experimental protocols
2.2.3 Optimisation of the model fit
2.3 Results
2.3.1 Burst length dependence
2.3.2 Frequency dependence
2.3.3 Safo and Regehr
2.3.4 Model comparison
2.3.5 Prediction: PF → LTD
2.4 Discussion
3 Maintenance of memory in a cortical network model 
3.1 Introduction
3.2 Materials and Methods
3.2.1 Calcium-based plasticity model
3.2.2 Probability density function of the calcium concentration
3.2.3 Diffusion approximation for the synaptic efficacy with a flat potential
3.2.4 Kramers expected escape time from a double-well potential
3.2.5 Numerical methods: Event-based implementation
3.2.6 The network
3.2.7 Numerical methods: Network simulations
3.2.8 Computing analytically mean firing rates and E-E synaptic efficacy
3.3 Results
3.3.1 Memory behaviour for a synapse connecting two independent Poisson neurons
3.3.2 Memory decay for a bistable synapse
3.3.3 Steady-state behaviour of networks of LIF neurons with plastic synapses
3.3.4 Memory decay in a recurrent network of LIF neurons
3.3.5 Memory induction in a recurrent network of LIF neurons
3.4 Discussion
4 Conclusion 
4.1 Cerebellar plasticity model
4.2 Hippocampal and neocortical plasticity model
4.3 Outlook
A Simulation parallelisation
B Generation of random numbers in parallel
C Shot noise
C.1 Mapping calcium dynamics to shot noise
C.2 Solving the shot noise equation

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