nips nips2002 nips2002-129 knowledge-graph by maker-knowledge-mining

129 nips-2002-Learning in Spiking Neural Assemblies

Source: pdf

Author: David Barber

Abstract: We consider a statistical framework for learning in a class of networks of spiking neurons. Our aim is to show how optimal local learning rules can be readily derived once the neural dynamics and desired functionality of the neural assembly have been speciﬁed, in contrast to other models which assume (sub-optimal) learning rules. Within this framework we derive local rules for learning temporal sequences in a model of spiking neurons and demonstrate its superior performance to correlation (Hebbian) based approaches. We further show how to include mechanisms such as synaptic depression and outline how the framework is readily extensible to learning in networks of highly complex spiking neurons. A stochastic quantal vesicle release mechanism is considered and implications on the complexity of learning discussed. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract We consider a statistical framework for learning in a class of networks of spiking neurons. [sent-6, score-0.304]

2 Our aim is to show how optimal local learning rules can be readily derived once the neural dynamics and desired functionality of the neural assembly have been speciﬁed, in contrast to other models which assume (sub-optimal) learning rules. [sent-7, score-0.852]

3 Within this framework we derive local rules for learning temporal sequences in a model of spiking neurons and demonstrate its superior performance to correlation (Hebbian) based approaches. [sent-8, score-0.508]

4 We further show how to include mechanisms such as synaptic depression and outline how the framework is readily extensible to learning in networks of highly complex spiking neurons. [sent-9, score-0.799]

5 A stochastic quantal vesicle release mechanism is considered and implications on the complexity of learning discussed. [sent-10, score-0.508]

6 1 Introduction Models of individual neurons range from simple rate based approaches to spiking models and further detailed descriptions of protein dynamics within the cell[9, 10, 13, 6, 12]. [sent-11, score-0.541]

7 As the experimental search for the neural correlates of memory increasingly consider multi-cell observations, theoretical models of distributed memory become more relevant[12]. [sent-12, score-0.157]

8 Despite increasing complexity of neural description, many theoretical models of learning are based on correlation Hebbian assumptions – that is, changes in synaptic eﬃcacy are related to correlations of preand post-synaptic ﬁring[9, 10, 14]. [sent-13, score-0.313]

9 Whilst such learning rules have some theoretical justiﬁcation in toy neural models, they are not necessarily optimal in more complex cases in which the dynamics of the cell contains historical information, such as modelled by synaptic facilitation and depression, for example[1]. [sent-14, score-0.69]

10 It is our belief that appropriate synaptic learning rules should appear as a natural consequence of the neurodynamical system and some desired functionality – such as storage of temporal sequences. [sent-15, score-0.489]

11 It seems clear that, as the brain operates dynamically through time, relevant cognitive processes are plausibly represented in vivo as temporal sequences of spikes in restricted neural assemblies. [sent-16, score-0.169]

12 This paradigm has heralded a new research front in dynamic systems of spiking neurons[10]. [sent-17, score-0.269]

13 However, to date, many learning algorithms assume Hebbian learning, and assess its performance in a given model[8, 6, 14]. [sent-18, score-0.051]

14 neuron j neuron i Highly Complex (deterministic) Internal Dynamics h(1) h(2) h(t) v(1) v(2) v(t) (a) Deterministic Hiddens stochastic firing axon . [sent-20, score-0.413]

15 (b) Neural ﬁring model Figure 1: (a) A ﬁrst order Dynamic Bayesian Network with deterministic hidden states (represented by diamonds). [sent-21, score-0.304]

16 Recent work[13] has taken into account some of the complexities in the synaptic dynamics, including facilitation and depression, and derived appropriate learning rules. [sent-23, score-0.291]

17 However, these are rate based models, and do not capture the detailed stochastic ﬁring eﬀects of individual neurons. [sent-24, score-0.124]

18 Other recent work [4] has used experimental observations to modify Hebbian learning rules to make heuristic rules consistent with empirical observations[11]. [sent-25, score-0.237]

19 However, as more and more detail of cellular processes are experimentally discovered, it would be satisfying to see learning mechanisms as naturally derivable consequences of the underlying cellular constraints. [sent-26, score-0.309]

20 This paper is a modest step in this direction, in which we outline a framework for learning in spiking systems which can handle highly complex cellular processes. [sent-27, score-0.379]

21 The major simplifying assumption is that internal cellular processes are deterministic, whilst communication between cells can be stochastic. [sent-28, score-0.197]

22 The central aim of this paper is to show that optimal learning algorithms are derivable consequences of statistical learning criteria. [sent-29, score-0.187]

23 2 A Framework for Learning A neural assembly of V neurons is represented by a vector v(t) whose components vi (t), i = 1, . [sent-31, score-0.533]

24 Throughout we assume that vi (t) ∈ {0, 1}, for which vi (t) = 1 means that neuron i spikes at time t, and vi (t) = 0 denotes no spike. [sent-35, score-0.833]

25 The shape of an action potential is assumed therefore not to carry any information. [sent-36, score-0.132]

26 This constraint of a binary state ﬁring representation could be readily relaxed without great inconvenience to multiple or even continuous states. [sent-37, score-0.106]

27 Our stated goal is to derive optimal learning rules for an assumed desired functionality and a given neural dynamics. [sent-38, score-0.281]

28 To make this more concrete, we assume that the task is sequence learning (although generalistions to other forms of learning, including input-output type dynamics are readily achievable[2]). [sent-39, score-0.361]

29 We make the important assumption that the neural assembly has a sequence of states V = {v(1), v(2), . [sent-40, score-0.344]

30 In addition to the neural ﬁring states, V, we assume that there are hidden/latent variables which inﬂuence the dynamics of the assembly, but which cannot be directly observed. [sent-44, score-0.253]

31 These might include protein levels within a cell, for example. [sent-45, score-0.059]

32 These variables may also represent environmental conditions external to the cell and common to groups of cells. [sent-46, score-0.12]

33 We represent a sequence of hidden variables by H = {h(1), h(2), . [sent-47, score-0.21]

34 The general form of our model is depicted in ﬁg(1)[a] and comprises two components 1. [sent-51, score-0.045]

35 The distribution is parameterised by θ v , which can be learned from a training sequence (see below). [sent-53, score-0.104]

36 Deterministic Hidden Variable Updating : h(t + 1) = f (v(t + 1), v(t), h(t), θ h ) (2) This equation speciﬁes that the next hidden state of the assembly h(t + 1) depends on a vector function f of the states v(t+1), v(t), h(t). [sent-57, score-0.389]

37 The function f is parameterised by θ h which is to be learned. [sent-58, score-0.044]

38 This model is a special case of Dynamic Bayesian networks, in which the hidden variables are deterministic functions of their parental states and is treated in more generality in [2]. [sent-59, score-0.342]

39 The model assumptions are depicted in ﬁg(1)[b] in which potentially complex deterministic interactions within a neuron can be considered, with lossy transmission of this information between neurons in the form of stochastic ﬁring. [sent-60, score-0.599]

40 Whilst the restriction to deterministic hidden dynamics appears severe, it has the critical advantage that learning in such models can be achieved by deterministic forward propagation through time. [sent-61, score-0.694]

41 The key mechanism for learning in statistical models is maximising the log-likelihood L(θ v , θ h |V) of a sequence V, T −1 L(θ v , θ h |V) = log p(v(1)|θ v ) + log p(v(t + 1)|v(t), h(t), θ v ) (3) t=1 where the hidden unit values are calculated recursively using (2). [sent-64, score-0.367]

42 P is straightforward using the log-likelihood µ µ L(θ v , θ h |V ). [sent-68, score-0.045]

43 In a biological system it is natural to use gradient ascent training θ ← θ +ηdL/dθ where the learning rate η is chosen small enough to ensure convergence to a local optimum of the likelihood. [sent-73, score-0.116]

44 This batch training procedure is readily convertible to an online form if needed. [sent-74, score-0.152]

45 Highly complex functions f and tables p(v(t + 1)|v(t), h(t)) may be used. [sent-76, score-0.073]

46 In the remaining sections, we apply this framework to some simple models and show how optimal learning rules can be derived for old and new theoretical models. [sent-77, score-0.215]

47 1 Stochastically Spiking Neurons We assume that neuron i ﬁres depending on the membrane potential ai (t) through p(vi (t + 1) = 1|v(t), h(t)) = p(vi (t + 1) = 1|ai (t)). [sent-79, score-0.795]

48 (More complex dependencies on environmental variables are also clearly possible). [sent-80, score-0.117]

49 The log-likelihood of a sequence of visible states V is T −1 V log σ ((2vi (t + 1) − 1)ai (t)) L= (8) t=1 i=1 and the (online) gradient of the log-likelihood is then dL(t + 1) dai (t) = (vi (t + 1) − σ(ai (t))) dwij dwij (9) where we used the fact that vi ∈ {0, 1}. [sent-84, score-0.774]

50 The batch gradient is simply given by summing the above online gradient over time. [sent-85, score-0.15]

51 Here wij are parameters of the membrane potential (see below). [sent-86, score-0.714]

52 We take (9) as common to the remainder in which we model the membrane potential ai (t) with increasing complexity. [sent-87, score-0.636]

53 2 A simple model of the membrane potential Perhaps the simplest membrane potential model is the Hopﬁeld potential V ai (t) ≡ wij vj (t) − bi (10) j=1 where wij characterizes the synaptic eﬃcacy from neuron j (pre-synaptic) to neuron i (post-synaptic), and bi is a threshold. [sent-89, score-2.581]

54 The potential is a deterministic function of the network state and (the collection of) membrane potentials inﬂuences the next state of the network. [sent-92, score-0.607]

55 (b) Dynamic synapses correspond to hidden variables which inﬂuence the membrane potential and update themselves, depending on the ﬁring of the network. [sent-93, score-0.659]

56 Only one membrane potential and one synaptic factor is shown. [sent-94, score-0.588]

57 Note that in the above rule vi (t + 1) refers to the desired known training pattern, and σ(ai (t)) can be interpreted as the average instantaneous ﬁring rate of neuron i at time t + 1 when its inputs are clamped to the known desired values of the network at time t. [sent-96, score-0.642]

58 The above learning rule can be seen as a modiﬁcation of the standard T −1 Hebb learning rule wij = t=1 vi (t + 1)vj (t). [sent-98, score-0.768]

59 However, the rule (11) can store a sequence of V linearly independent patterns, much greater than the 0. [sent-99, score-0.16]

60 Biologically, the rule (11) could be implemented by measuring the diﬀerence between the desired training state vi (t + 1) of neuron i, and the instantaneous ﬁring rate of neuron i when all other neurons, j = i are clamped in training states vj (t). [sent-101, score-1.053]

61 3 Dynamic Synapses In more realistic synaptic models, neurotransmitter generation depends on a ﬁnite rate of cell subcomponent production, and the quantity of vesicles released is affected by the history of ﬁring[1]. [sent-103, score-0.33]

62 The depression mechanism aﬀects the impact of spiking on the membrane potential response by moderating terms in the membrane potential ai (t) of the form j wij vj (t) to j wij xj (t)vj (t), for depression factors xj (t) ∈ [0, 1]. [sent-104, score-2.639]

63 where δt, τ , and U represent time scales, recovery times and spiking eﬀect parameters respectively. [sent-108, score-0.181]

64 Note that these depression factor dynamics are exactly of the form of hidden variables that are not observed, consistent with our framework in section (2), see ﬁg(2)[b]. [sent-109, score-0.521]

65 Whilst some previous models have considered learning rules for dynamic synapses using spiking-rate models [13, 15] we consider learning in a stochastic spiking model. [sent-110, score-0.741]

66 Also, in contrast to a previous study which assumes that the synaptic dynamics modulates baseline Hebbian weights[14], we show below that it is straightforward to include dynamic synapses in a principled way using our learning framework. [sent-111, score-0.681]

67 Since the depression dynamics in this model do not explicitly depend on wij , the gradients are simple to calculate. [sent-112, score-0.649]

68 Note that synaptic facilitation is also straightforward to include in principle[15]. [sent-113, score-0.315]

69 For the Hopﬁeld potential, the learning dynamics is simply given by equations (9,12), with dai (t) = xj (t)vj (t). [sent-114, score-0.409]

70 In ﬁg(3) we demonstrate learning a random temdwij poral sequence of 20 time steps for an assembly of 50 neurons. [sent-115, score-0.304]

71 After learning w ij with our rule, we initialised the trained network in the ﬁrst state of the training sequence. [sent-116, score-0.083]

72 The remaining states of the sequence were then correctly recalled by iteration of the learned model. [sent-117, score-0.112]

73 For comparison, we plot the results of using the dynamics having set the w ij using a temporal Hebb rule. [sent-119, score-0.238]

74 The poor performance of the correlation based Hebb rule demonstrates the necessity, in general, to couple a dynamical system with an appropriate learning mechanism which, in this case at least, is readily available. [sent-120, score-0.244]

75 4 Leaky Integrate and Fire models Leaky integrate and ﬁre models move a step towards biological realism in which the membrane potential increments if it receives an excitatory stimulus (wij > 0), and decrements if it receives an inhibitory stimulus (wij < 0). [sent-121, score-0.519]

76 A model that incorporates such eﬀects is   ai (t) = αai (t − 1) + j wij vj (t) + θ rest (1 − α) (1 − vi (t − 1)) + vi (t − 1)θ f ired (13) Since vi ∈ {0, 1}, if neuron i ﬁres at time t − 1 the potential is reset to θ f ired at time t. [sent-122, score-1.889]

77 Similarly, with no synaptic input, the potential equilibrates to θ rest with time constant −1/ log α. [sent-123, score-0.376]

78 Here α ∈ [0, 1] represents membrane leakage characteristic of this class of models. [sent-124, score-0.271]

79 a(t − 1) a(t) a(t + 1) r(t − 1) r(t) r(t + 1) v(t − 1) v(t) v(t + 1) Figure 4: Stochastic vesicle release (synaptic dynamic factors not indicated). [sent-125, score-0.392]

80 Despite the apparent increase in complexity of the membrane potential over the simple Hopﬁeld case, deriving appropriate learning dynamics for this new system is straightforward since, as before, the hidden variables (here the membrane potentials) update in a deterministic fashion. [sent-126, score-1.236]

81 The membrane derivatives are dai (t) dai (t − 1) = (1 − vi (t − 1)) α + vj (t) dwij dwij (14) i (t=1) By initialising the derivative dadwij = 0, equations (9,13,14) deﬁne a ﬁrst order recursion for the gradient which can be used to adapt wij in the usual manner wij ← wij + ηdL/dwij . [sent-127, score-2.152]

82 We could also apply synaptic dynamics to this case by replacing the term vj (t) in (14) by xj (t)vj (t). [sent-128, score-0.666]

83 5 A Stochastic Vesicle Release Model Neurotransmitter release can be highly stochastic and it would be desirable to include this mechanism in our models. [sent-130, score-0.305]

84 The membrane potential is then governed in integrate and ﬁre models by   ai (t) = αai (t − 1) + j wij rij (t) + θ rest (1 − α) (1 − vi (t − 1)) + vi (t − 1)θ f ired (17) This model is schematically depicted in ﬁg(4). [sent-132, score-1.753]

85 Since the unobserved stochastic release variables rij (t) are hidden, this model does not have fully deterministic hidden dynamics. [sent-133, score-0.691]

86 In general, learning in such models is more complex and would require both forward and backward temporal propagations including, undoubtably, graphical model approximation techniques[7]. [sent-134, score-0.232]

87 6 Discussion Leaving aside the issue of stochastic vesicle release, a further step in the evolution of membrane complexity is to use Hodgkin-Huxley type dynamics[9]. [sent-135, score-0.504]

88 Whilst this might appear complex, in principle, this is straightforward since the membrane dynamics can be represented by deterministic hidden dynamics. [sent-136, score-0.744]

89 Explicitly summing out the hidden variables would then give a representation of Hodgkin-Huxley dynamics analogous to that of the Spike Response Model (see Gerstner in [10]). [sent-137, score-0.326]

90 Deriving optimal learning in assemblies of stochastic spiking neurons can be achieved using maximum likelihood. [sent-138, score-0.463]

91 This is straightforward in cases for which the latent dynamics is deterministic. [sent-139, score-0.221]

92 It is worth emphasising, therefore, that almost arbitrarily complex spatio-temporal patterns may potentially be learned – and generated under cued retrieval – for very complex neural dynamics. [sent-140, score-0.155]

93 Whilst this framework cannot deal with arbitrarily complex stochastic interactions, it can deal with learning in a class of interesting neural models, and concepts from graphical models can be useful in this area. [sent-141, score-0.3]

94 A more general stochastic framework would need to examine approximate causal learning rules which, despite not being fully optimal, may perform well. [sent-142, score-0.272]

95 Finally, our assumption that the brain operates optimally (albeit within severe constraints) enables us to drop other assumptions about unobserved processes, and leads to models with potentially more predictive power. [sent-143, score-0.137]

96 Nelson, Synaptic depression and cortical gain control, Science 275 (1997), 220–223. [sent-151, score-0.162]

97 Agakov, Correlated sequence learning in a network of spiking neurons using maximum likelihood, Tech. [sent-156, score-0.38]

98 Sherrington, Phase diagram and storage capacity u of sequence processing neural networks, Journal of Physics A 31 (1998), 8607–8621. [sent-168, score-0.099]

99 van Hemmen, Hebbian learning and spiking neurons, Physical Review E 59 (1999), 4498–4514. [sent-182, score-0.232]

100 Sakmann, Regulation of synaptic eﬃcacy by coindence of postsynaptic APs and EPSPs, Science 275 (1997), 213–215. [sent-192, score-0.185]

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