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Hippocampal Spike Timing–Dependent Plasticity and Phase Response Curves

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Hippocampal Spike Timing–Dependent Plasticity and Phase Response Curves COMPUTATION AND SYSTEMS ARTICLES Matching storage and recall: hippocampal spike timing–dependent plasticity and phase response curves Ma´te´ Lengyel1, Jeehyun Kwag2, Ole Paulsen2 & Peter Dayan1 Hippocampal area CA3 is widely considered to function as an autoassociative memory. However, it is insufficiently understood how it does so. In particular, the extensive experimental evidence for the importance of carefully regulated spiking times poses the question as to how spike timing–based dynamics may support memory functions. Here, we develop a normative theory of autoassociative memory encompassing such network dynamics. Our theory specifies the way that the synaptic plasticity rule of a memory constrains the form of neuronal interactions that will retrieve memories optimally. If memories are stored by spike timing– http://www.nature.com/natureneuroscience dependent plasticity, neuronal interactions should be formalized in terms of a phase response curve, indicating the effect of presynaptic spikes on the timing of postsynaptic spikes. We show through simulation that such memories are competent analog autoassociators and demonstrate directly that the attributes of phase response curves of CA3 pyramidal cells recorded in vitro qualitatively conform with the theory. The task of storing memories and recalling them from partial or noisy analysis methods have wide application in everything from the analysis cues is fundamental for the brain and has been a particular focus for of cardiac rhythms28 and patterns of oscillatory coordination for motor empirical1,2 and theoretical work3 on the hippocampus. This naturally pattern generation29 to the relationship between nonlinear and sub- raises the key question as to how the properties of single cells and the threshold intrinsic mechanisms within cells and various forms of overall hippocampal network support its proposed function. The CA3 synchrony30,31. However, to our knowledge, hitherto they have not region of the hippocampus has the densest recurrent collateral system been used to characterize oscillatory autoassociative memories. 4 2005 Nature Publishing Group Group 2005 Nature Publishing in the mammalian cortex , which is consistent with the central role Here, we present a theory treating autoassociative recall as optimal 32,33 © accorded to recurrent connections in standard models of autoassocia- probabilistic inference , inferring the recurrent dynamics within a tive memories5. However, with relatively few exceptions (in the memory that are normatively matched to the form of the synaptic hippocampus and elsewhere6–13), such models typically use a highly plasticity rule used to store traces. In the case of CA3, this encompasses simplified treatment of the resulting collective dynamics of their model memories encoded in spike timings relative to underlying neural neurons14,15. They thereby fail to capture a salient characteristic of the oscillations, and it thus involves inferring the optimal PRC of neurons activity of hippocampal neurons during memory states: namely, the from their spike timing–dependent synaptic plasticity (STDP) rule. We central role played by spike timing. thus make specific predictions about the shape of the PRC of CA3 Evidence for the importance of timing in the hippocampus is pyramidal neurons based on the STDP reported in cultured hippo- widespread. Behaviorally relevant neural oscillations at different fre- campal neurons34. We show that the theoretically derived PRC provides quencies pace the activity of all hippocampal cell types. In addition, the a good qualitative match to those recorded in hippocampal CA3 timing of individual action potentials of principal cells is tightly pyramidal cells in vitro. regulated16,17, and temporal sequences of the ensemble firing pattern consistently reappear during both awake behavior18,19 and sleep20–22. RESULTS Furthermore, synaptic plasticity is also critically sensitive to the precise Autoassociative recall as probabilistic inference timing of pre- and postsynaptic spikes23,24. The fundamental requirement for an autoassociative memory is to A key idea for understanding networks such as CA3 in which recall a previously encoded memory trace when cued with a noisy or information may be coded by spike times is the phase response curve partial cue. As synaptic plasticity, the key mechanism for long term (PRC). PRCs offer a precise characterization of the effect a presynaptic storage in the brain, loses information about the traces, recall poses a spike has on the timing of the succeeding postsynaptic spike depending complex problem of probabilistic inference. The optimal solution to on its time of arrival25–27. PRCs and related phase reduction and this problem, which amounts to a normative theory of recall, depends 1Gatsby Computational Neuroscience Unit, University College London, 17 Queen Square, London WC1N 3AR, UK. 2University Laboratory of Physiology, Oxford University, Parks Road, Oxford OX1 3PT, UK. Correspondence should be addressed to M.L. ([email protected]). Received 19 April; accepted 12 September; published online 30 October 2005; doi:10.1038/nn1561 NATURE NEUROSCIENCE VOLUME 8 [ NUMBER 12 [ DECEMBER 2005 1677 ARTICLES contribution. Indeed, equation (2) is noninvertible, with many differ- abSpike timing–dependent plasticity i =1 234 N ent combinations of synaptic weight changes potentially leading to the 1 same total synaptic weight. This lossiness implies that recall from a noisy or partial cue involves a process of combining probabilistic 0 information from (i) the general statistical properties of the traces W X X X X X 1N 1 2 3 4 N (that is, the prior distribution), (ii) the cue itself and (iii) the synaptic W 13 weights. Finding the statistically most likely memory trace involves W W W –1 12 32 N4 change Synaptic weight –500 50 complex, nonlocal operations. However, it can be well approximated t – t (ms) pre post (Supplementary Note online) by a form of neural dynamics among the Coupling function Phase response curves recurrently coupled neurons in which there are explicit terms corre- c 2 d 0.2 sponding exactly to each of these three sources of information. 1 Under this account, synaptic interactions between neurons of the Delay 0.1 network implement a ‘search’ for the activity pattern associated with 0 0 the original memory trace that was most likely to have led to the cue. Advance –0.1 Over the course of search, each neuron gradually changes its spike Phase response (rad) –1 Phase response (rad) –2 0 2 –2 0 2 φ φ timing such that the activities of it and its synaptic partners are pre– post (rad) Phase of EPSP (rad) increasingly likely to reflect a pattern consistent with the corresponding Figure 1 Normative theory of spike timing–based autoassociative memory. synaptic weights. As a result, the contribution to the dynamics of (a) Schematic diagram of a recurrent network of neurons. Neurons are neuron i associated with the synaptic weight term involves a linear sum numbered i ¼ 1 y N and are characterized by their respective activities, of influences from its presynaptic afferents j, with each element in the y x1 xN. Presynaptic neuron j is connected to postsynaptic neuron i through sum taking the form a recurrent synapse with efficacy (weight) wij. Although all-to-all connectivity (excluding autapses) was assumed for the theoretical derivations, here only @ Hðx ; x Þ¼w Oðx ; x Þð3Þ a few synapses are shown for clarity. (b) Memories are stored by a spike i j ij @x i j http://www.nature.com/natureneuroscience timing-dependent plasticity (STDP) rule derived from experiments on i 34 cultured hippocampal neurons . tpre and tpost represent times of pre- and This has the obvious appealing characteristic that the strength of the postsynaptic firing. Gray lines are exponential fits24 to data from ref. 34. interaction between two neurons is scaled by the synaptic weight. Less Black line is a continuous fit taken to be the synaptic learning rule (O)in obvious is our key suggestion that the interaction should be determined equation (1). (c) Optimal coupling function (H) for retrieving memories stored by STDP (black line in b), as derived in equation (3). This shows how the by the derivative of the synaptic plasticity rule that was used to store firing phase of the postsynaptic neuron should change as a function of the memories in the network. We can derive an intuition about this rule by phase difference between pre- and postsynaptic firing, if neurons were to considering what happens if the synaptic weight is positive; that is, interact continuously. fpre and fpost represent firing phases of pre- and above an average value. This large synaptic weight arises from the postsynaptic cells relative to a local field potential oscillation. (d) Optimal weight changes associated with the memory traces. Therefore, the phase response curves (PRCs) derived from the optimal coupling function neuron should change its activity to increase the contribution that it (shown in c), showing how neurons should interact through spikes. Different and its synaptic partner would have made to the synaptic weight had curves correspond to linearly increasing synaptic weights (in increasing order: red, yellow, green, blue). ‘Zero’ phase is the phase of the postsynaptic spike. their present activity indeed been associated with one of the memories 2005 Nature Publishing Group Group 2005 Nature Publishing that caused this excess synaptic weight. © Here, we study the case of area CA3 in the hippocampus, in which on the statistical characteristics of the traces and cues and the nature of the synaptic plasticity rule involves STDP. We show that the optimal the lossiness of storage. dynamics for neurons representing memory traces in terms of the More concretely, consider a network of N neurons (Fig. 1a), fully phase of firing relative to an underlying oscillation is determined by a connected by N Â (N – 1) synaptic connections, and storing particular shape of PRC that we experimentally validate. M memory traces. Memories are represented by distributed patterns m of neural activity, with scalar variable xi characterizing the ith neuron Spike timing-based memories m in the mth memory trace.
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