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197 nips-2006-Uncertainty, phase and oscillatory hippocampal recall

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Author: Máté Lengyel, Peter Dayan

Abstract: Many neural areas, notably, the hippocampus, show structured, dynamical, population behavior such as coordinated oscillations. It has long been observed that such oscillations provide a substrate for representing analog information in the ﬁring phases of neurons relative to the underlying population rhythm. However, it has become increasingly clear that it is essential for neural populations to represent uncertainty about the information they capture, and the substantial recent work on neural codes for uncertainty has omitted any analysis of oscillatory systems. Here, we observe that, since neurons in an oscillatory network need not only ﬁre once in each cycle (or even at all), uncertainty about the analog quantities each neuron represents by its ﬁring phase might naturally be reported through the degree of concentration of the spikes that it ﬁres. We apply this theory to memory in a model of oscillatory associative recall in hippocampal area CA3. Although it is not well treated in the literature, representing and manipulating uncertainty is fundamental to competent memory; our theory enables us to view CA3 as an effective uncertainty-aware, retrieval system. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Uncertainty, phase and oscillatory hippocampal recall M´ t´ Lengyel and Peter Dayan ae Gatsby Computational Neuroscience Unit University College London 17 Queen Square, London WC1N 3AR, United Kingdom {lmate,dayan}@gatsby. [sent-1, score-0.614]

2 It has long been observed that such oscillations provide a substrate for representing analog information in the ﬁring phases of neurons relative to the underlying population rhythm. [sent-5, score-0.768]

3 However, it has become increasingly clear that it is essential for neural populations to represent uncertainty about the information they capture, and the substantial recent work on neural codes for uncertainty has omitted any analysis of oscillatory systems. [sent-6, score-0.39]

4 Here, we observe that, since neurons in an oscillatory network need not only ﬁre once in each cycle (or even at all), uncertainty about the analog quantities each neuron represents by its ﬁring phase might naturally be reported through the degree of concentration of the spikes that it ﬁres. [sent-7, score-1.712]

5 We apply this theory to memory in a model of oscillatory associative recall in hippocampal area CA3. [sent-8, score-0.616]

6 Although it is not well treated in the literature, representing and manipulating uncertainty is fundamental to competent memory; our theory enables us to view CA3 as an effective uncertainty-aware, retrieval system. [sent-9, score-0.337]

7 Most groups working on the theory of spiking oscillatory networks have considered only the second of these – this is true, for instance, of Hopﬁeld’s work on olfactory representations [2] and Yoshioka’s [3] and Lengyel & Dayan’s work [4] on analog associative memories in CA3. [sent-11, score-0.433]

8 Since neurons do really ﬁre more or less than one spike per cycle, and furthermore in a way that can be informationally rich [5, 6], this poses a key question as to what the other dimensions convey. [sent-12, score-0.448]

9 The number of spikes per cycle is an obvious analog of a conventional ﬁring rate. [sent-13, score-0.407]

10 Single neurons can convey the certainty of a binary proposition by ﬁring more or less strongly [10, 11]; a whole population can use ﬁring rates to convey uncertainty about a collectively-coded analog quantity [12]. [sent-15, score-0.911]

11 However, if neurons can ﬁre multiple spikes per cycle, then the degree to which the spikes are concentrated around a mean phase is an additional channel for representing information. [sent-16, score-1.15]

12 Concentration is not merely an abstract quantity; rather we can expect that the effect of the neuron on its postsynaptic partners will be strongly inﬂuenced by the burstiness of the spikes, an effect apparent, for instance, in the complex time-courses of short term synaptic dynamics. [sent-17, score-0.303]

13 Here, we suggest that concentration codes for the uncertainty about phase – highly concentrated spiking represents high certainty about the mean phase in the cycle. [sent-18, score-1.049]

14 One might wonder whether uncertainty is actually important for the cases of oscillatory processing that have been identiﬁed. [sent-19, score-0.245]

15 One key computation for spiking oscillatory networks is memory retrieval [3, 4]. [sent-20, score-0.534]

16 Although it is not often viewed this way, memory retrieval is a genuinely probabilistic task [13, 14], with the complete answer to a retrieval query not being a single memory pattern, but rather a distribution over memory patterns. [sent-21, score-0.961]

17 This is because at the time of the query the memory device only has access to incomplete information regarding the memory trace that needs to be recalled. [sent-22, score-0.522]

18 Most importantly, the way memory traces are stored in the synaptic weight matrix implies a data lossy compression algorithm, and therefore the original patterns cannot be decompressed at retrieval with absolute certainty. [sent-23, score-0.616]

19 In this paper, we ﬁrst describe how oscillatory structures can use all three activity characteristics at their disposal to represent two pieces of information and two forms of uncertainty (Section 2). [sent-24, score-0.34]

20 We then suggest that this representational scheme is appropriate as a model of uncertainty-aware probabilistic recall in CA3. [sent-25, score-0.166]

21 We show in numerical simulations that the derived dynamics lead to competent memory retrieval, supplemented by uncertainty signals that are predictive of retrieval errors (Section 4). [sent-27, score-0.65]

22 2 Representation Single cell The heart of our proposal is a suggestion for how to interpret the activity of a single neuron in a single oscillatory cycle (such as a theta-cycle in the hippocampus) as representing a probability distribution. [sent-28, score-0.606]

23 We treat the implied distribution over the true phase x as being conditional on z. [sent-32, score-0.375]

24 However, if z = 1, then the distribution over x is a mixture of q (x), a uniform distribution on [0, T ), and a narrow, quasi-delta, distribution q⊥ (x; φ) (of width T ) around the mean ﬁring phase (φ) of the spikes. [sent-34, score-0.316]

25 The natural alternative is to consider an approximation in which neurons make independent contributions, with marginals as in equation 2. [sent-39, score-0.391]

26 A) A neuron’s ﬁring times during a period [0, T ) are described by three parameters: r, the number of spikes; φ the mean phase of those spikes; and c, the phase concentration. [sent-41, score-0.632]

27 C) If z = 1, then φ and c jointly deﬁne a distribution over phase which is a mixture (weighted by γ(c)) of a distribution peaked at φ and a uniform distribution. [sent-43, score-0.316]

28 Dynamics When the actual distribution P the population has to represent lies outside the class of representable distributions Q in equation 3 with independent marginals, a key computational step is to ﬁnd activity parameters φ, c, r for the neurons that make Q as close to P as possible. [sent-45, score-0.58]

29 1 We have thus suggested a general representational framework, in which the speciﬁcation of a computational task amounts to deﬁning a P distribution which the network should represent as best as possible. [sent-48, score-0.167]

30 Equation 5 then deﬁnes the dynamics of the interaction between the neurons that optimizes the network’s approximation. [sent-49, score-0.462]

31 3 CA3 memory One of the most widely considered tasks that recurrent neural networks need to solve is that of autoassociative memory storage and retrieval. [sent-50, score-0.633]

32 Moreover, hippocampal area CA3, which is thought to play a key role in memory processing, exhibits oscillatory dynamics in which ﬁring phases are known to play an important functional role. [sent-51, score-0.762]

33 We characterize the activity in CA3 neurons during recall as representing the probability distribution over memories being recalled. [sent-53, score-0.771]

34 Treating storage from a statistical perspective, we use Bayes rule to deﬁne a posterior distribution over the memory pattern implied by a noisy and impartial cue. [sent-54, score-0.384]

35 This distribution is represented approximately by the activities φi , ri , ci of the neurons in the network as in equation 3. [sent-55, score-0.521]

36 Recurrent dynamics among the neurons as in equation 5 ﬁnd appropriate values of these parameters, and model network interactions during recall in CA3. [sent-56, score-0.696]

37 a ﬁring phase (xi ∈ [0, T ), where T is the period of the population oscillation. [sent-59, score-0.383]

38 Posterior for memory recall Following [14, 4], we characterize retrieval in terms of the posterior distribution over x, z given three sources of information: a recall cue (˜, ˜), the synaptic weight max z trix, and the prior over the memories. [sent-61, score-0.751]

39 Dynamics for memory recall Plugging the posterior from equation 8 to the general dynamics equation 5 yields the neuronal update rules that will be appropriate for uncertainty-aware memory recall, and which we treat as a model of recurrent dynamics in CA3. [sent-67, score-1.028]

40 These dynamics generalize, and thus inherit, some of the characteristics of the purely phase-based network suggested in [4]. [sent-70, score-0.206]

41 This means that they also inherit the match with physiologically-measured phase response curves (PRCs) from in vitro CA3 neurons that were measured to test this suggestion [16]. [sent-71, score-0.825]

42 The key difference here is that we expect the magnitude (though not the shape) of the inﬂuence of a presynaptic neuron on the phase of a postsynaptic one to scale with its rate, for high concentration. [sent-72, score-0.577]

43 Preliminary in vitro results show that PRCs recorded in response to burst stimulation are not qualitatively different from PRCs induced by single spikes; however, it remains to be seen if their magnitude scales in the way implied by the dynamics here. [sent-73, score-0.343]

44 Time evolution of ﬁring phases (left panels), concentrations (middle panels), and rates (right panels) of neurons that should (top row) or should not (bottom row) participate in the memory pattern being retrieved. [sent-80, score-0.963]

45 Note that ﬁring phases in the top row are plotted as a difference from the stored ﬁring phases so that φ = 0 means perfect retrieval. [sent-81, score-0.421]

46 Color code shows precision (blue: low, yellow: high) of the phase of the input to neurons, with red lines showing cells receving incorrect input rate. [sent-82, score-0.352]

47 4 Simulations Figure 2 shows the course of recall in the full network (with N = 100 neurons, and 10 stored patterns with pz = 0. [sent-83, score-0.448]

48 The top left panel shows that neurons that should ﬁre in the memory trace (ie for which z = 1) quickly converge on their correct phase, and that this convergence usually takes a longer time for neurons receiving more uncertain input. [sent-86, score-1.006]

49 This is paralleled by the way their ﬁring concentrations change (top middle panel): neurons with reliable input immediately increase their concentrations from the initial γ(c) = 0. [sent-87, score-0.498]

50 5 value to γ(c) = 1, while for those having more unreliable input it takes a longer time to build up conﬁdence about their ﬁring phases (and by the time they become conﬁdent their phases are indeed correct). [sent-88, score-0.365]

51 Finally, since the ﬁring rate input to the network is correct 90%, most neurons that should or should not ﬁre do or do not ﬁre, respectively, with maximal certainty about their rate (top and bottom right panels). [sent-90, score-0.672]

52 In particular, we may expect there to be a relationship between the actual error in the phase of ﬁring of the neurons recalled by the memory, and the ﬁring rates and concentrations (in the form of burst strengths) of the associated neurons themselves. [sent-92, score-1.434]

53 Here, we have sorted the neurons according to their burst strengths λ, and plotted histograms of errors in ﬁring phase for each group. [sent-95, score-0.82]

54 The lower the burst strength, the more likely are large errors – at least to an approximation. [sent-96, score-0.148]

55 A similar relationship exists between recalled (analogue) and stored (binary) ﬁring rates, where extreme values of the recalled ﬁring rate indicate that the stored ﬁring rate was 0 or 1 with higher certainty (Figure 3B). [sent-97, score-0.652]

56 He recorded neurons in hippocampal area CA1 (not CA3, although we may hope for some similar properties) whilst rats were shuttling on a linear track for food reward. [sent-99, score-0.461]

57 CA1 neurons have place ﬁelds – locations in the environment where they respond with spikes – and the phases of these spikes relative to the ongoing theta oscillation in the hippocampus are also known to convey information about location in space [5]. [sent-100, score-1.171]

58 To create the plot, we ﬁrst selected epochs with highquality and high power theta activity in the hippocampus (to ensure that phase is well estimated). [sent-101, score-0.578]

59 We then computed the mean ﬁring phase within the theta cycle, φ, of each neuron as a function of the location of the rat, separately for each visit to the same location. [sent-102, score-0.572]

60 We assumed that the ‘true’ phase x a neuron should recall at a given location is the average of these phases across different visits. [sent-103, score-0.759]

61 7 Stored firing rate A 0 Error in firing phase " 0 0. [sent-135, score-0.527]

62 " 0 ‘Error’ in firing phase " Figure 3: Uncertainty signals are predictive of the error a cell is making both in simulation (A,B), and as recorded from behaving animals (C). [sent-140, score-0.432]

63 Burst strength signals overall uncertainty about and thus predicts error in mean ﬁring phase (A,C), while graded ﬁring rates signal certainty about whether to ﬁre or not (B). [sent-141, score-0.667]

64 then evaluated the error a neuron was making at a given location on a given visit as the difference between its φ in that trial at that location and the ‘true’ phase x associated with that location. [sent-142, score-0.522]

65 This allowed us to compute statistics of the error in phase as a function of the burst strength. [sent-143, score-0.464]

66 The curves in the ﬁgure show that, as for the simulation, burst strength is at least partly inversely correlated with actual phase error, deﬁned in terms of the overall activity in the population. [sent-144, score-0.643]

67 One further way to evaluate the memory is to compare it to two existing associative memories that have previously been studied, and can be seen as special cases. [sent-146, score-0.452]

68 On one hand, our memory adds the dimension of phase to the uncertainty-aware rate-based memory that Sommer & Dayan [14] studied. [sent-147, score-0.838]

69 This memory made a somewhat similar variational approximation, but, as for the meanﬁeld Boltzmann machine [17], only involving r and ρ(r) and no phases. [sent-148, score-0.261]

70 On the other hand, the memory device can be seen as adding the dimension of rate to the phase-based memory that Lengyel & Dayan [4] treated. [sent-149, score-0.563]

71 Given these roots, we can follow the logic in ﬁgure 4 and compare the performance of our memory with these precursors in the cases for which they are designed. [sent-152, score-0.261]

72 For instance, to compare with the rate-based network, we construct memories which include phase information. [sent-153, score-0.45]

73 During recall, we present cues with relatively accurate rates, but relatively inaccurate phases, and evaluate the extent to which the network is perturbed by the presence of the phases (which, of course, it has to store in the single set of synaptic weights). [sent-154, score-0.397]

74 Here, a relatively small network (N = 100) was used, with memories that are dense (pz = 0. [sent-156, score-0.234]

75 Performance is evaluated by calculating the average error made in recalled ﬁring rates). [sent-158, score-0.135]

76 In the ﬁgure, the two blue curves are for the full model (with the phase information in the input being relatively unreliable, its circular concentration parameter distributed uniformly between 0. [sent-159, score-0.502]

77 1 and 10 across cells); the two yellow curves are for a network with only rates (which is similar to that described, but not simulated, by Sommer & Dayan [14]). [sent-160, score-0.228]

78 Exactly the same rate information is provided to all networks, and is 10% inaccurate (a degree known to the dynamics in the form of η0 and η1 ). [sent-161, score-0.147]

79 The two ﬂat dashed lines show the performance in the case that there are no recurrent synaptic weights at all. [sent-162, score-0.205]

80 This shows that the phase information, and the existence of phase uncertainty and processing during recall, does not A B 0. [sent-165, score-0.749]

81 coded model full model w/o learning full model 0. [sent-177, score-0.121]

82 1 1 10 100 Number of stored patterns 1000 1 10 100 Number of stored patterns 1000 Figure 4: Recall performance compared with a rate-only network (A) and a phase-only network (B). [sent-187, score-0.44]

83 Given its small size, the network is quite competent as an auto-associator. [sent-191, score-0.177]

84 Figure 4B shows a similar comparison between this network and a network that only has to deal with uncertainty in ﬁring phases but not in rates. [sent-192, score-0.486]

85 Again, its performance at recalling phase, given uncertain and noisy phase cues, but good rate-cues, is exactly on a par with the pure, phase-based network. [sent-193, score-0.404]

86 Further, the average errors are only modest, so the capacity of the network for storing analog phases is also impressive. [sent-194, score-0.388]

87 5 Discussion We have considered an interpretation of the activities of neurons in oscillating structures such as area CA3 of the hippocampus as representing distributions over two underlying quantities, one binary and one analogue. [sent-195, score-0.591]

88 We also showed how this representational capacity can be used to excellent effect in the key, uncertainty-sensitive computation of memory recall, an operation in which CA3 is known to be involved. [sent-196, score-0.328]

89 The resulting network model of CA3 encompasses critical aspects of its physiological properties, notably information-bearing ﬁring rates and phases. [sent-197, score-0.152]

90 Further, since it generalizes earlier theories of purely phase-based memories, this model is also consistent with the measured phase response curves of CA3 neurons, which characterize their actual dynamical interactions. [sent-198, score-0.385]

91 In vitro experiments along the lines of those carried out before [16], in which we have precise experimental control over pre- and post-synaptic activity can be used to test these predictions. [sent-201, score-0.189]

92 Further, making the sort of assumptions that underlie ﬁgure 3C, we can use data from awake behaving rats to see if the gross statistics of the changes in the activity of the neurons ﬁt the expectations licensed by the theory. [sent-202, score-0.516]

93 From a computational perspective, we have demonstrated that the network is a highly competent associative memory, correctly recalling both binary and analog information, along with certainty about it, and degrading gracefully in the face of overload. [sent-203, score-0.481]

94 In fact, compared with the representation of other analogue quantities (such as the orientation of a visually preseted bar), analogue memory actually poses a particularly tough problem for the representation of uncertainty. [sent-204, score-0.461]

95 By contrast, for analogue memory, each neuron has an independent analogue value, and so the dimensionality of the distribution scales with the number of neurons involved. [sent-206, score-0.701]

96 This extra representational power comes from the ability of neurons to distribute their spikes within a cycle to indicate their uncertainty about phase (using the dimension of time in just the same way that distributional population codes [12] used the dimension of neural space). [sent-207, score-1.273]

97 This dimension for representing analogue uncertainty is coupled to that of the ﬁring rate for representing binary uncertainty, since neurons have to ﬁre multiple times in a cycle to have a measurable lack of concentration. [sent-208, score-0.83]

98 However, this coupling is exactly appropriate given the form of the distribution assumed in equation 2, since weakly ﬁring neurons express only weak certainty about phase in any case. [sent-209, score-0.868]

99 In fact, it is conceivable that we could combine a different model for the ﬁring rate uncertainty with this model for analogue uncertainty, if, for instance, it is found that neuronal ﬁring rates covary in ways that are not anticipated from equation 2. [sent-210, score-0.33]

100 Exactly this seems to characterize the interaction between the hippocampus and the necortex during both consolidation and retrieval [18, 19]. [sent-212, score-0.239]

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