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162 nips-2006-Predicting spike times from subthreshold dynamics of a neuron

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Author: Ryota Kobayashi, Shigeru Shinomoto

Abstract: It has been established that a neuron reproduces highly precise spike response to identical ﬂuctuating input currents. We wish to accurately predict the ﬁring times of a given neuron for any input current. For this purpose we adopt a model that mimics the dynamics of the membrane potential, and then take a cue from its dynamics for predicting the spike occurrence for a novel input current. It is found that the prediction is signiﬁcantly improved by observing the state space of the membrane potential and its time derivative(s) in advance of a possible spike, in comparison to simply thresholding an instantaneous value of the estimated potential. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Predicting spike times from subthreshold dynamics of a neuron Ryota Kobayashi Department of Physics Kyoto University Kyoto 606-8502, Japan kobayashi@ton. [sent-1, score-1.057]

2 jp Abstract It has been established that a neuron reproduces highly precise spike response to identical ﬂuctuating input currents. [sent-8, score-0.894]

3 We wish to accurately predict the ﬁring times of a given neuron for any input current. [sent-9, score-0.29]

4 For this purpose we adopt a model that mimics the dynamics of the membrane potential, and then take a cue from its dynamics for predicting the spike occurrence for a novel input current. [sent-10, score-1.36]

5 It is found that the prediction is signiﬁcantly improved by observing the state space of the membrane potential and its time derivative(s) in advance of a possible spike, in comparison to simply thresholding an instantaneous value of the estimated potential. [sent-11, score-0.89]

6 In the simpliﬁcation proposed by FitzHugh [2] and Nagumo et al [3], the number of equations is reduced to two: the fast and slow variables which minimally represent the excitable dynamics. [sent-13, score-0.138]

7 The leaky integrate-and-ﬁre model [4], originally proposed far in advance of the Hodgkin-Huxley model, consists of only one variable that corresponds to the membrane potential, with a voltage resetting mechanism. [sent-14, score-0.48]

8 Those simpliﬁed models have been successful in not only extracting the essence of the dynamics, but also in reducing the computational cost of studying the large-scale dynamics of an assembly of neurons. [sent-15, score-0.133]

9 In contrast to such taste for simpliﬁcation, there are also a number of studies that pursue realism by developing multi-compartment models and installing newly found ionic channels. [sent-16, score-0.071]

10 User-friendly simulation platforms, such as NEURON [5] and GENESIS [6], enable experimental neurophysiologists to reproduce casually their experimental results or to explore potentially interesting phenomena for a new experiment to be performed. [sent-17, score-0.095]

11 Though those simulators have been successful in reproducing qualitative aspects of neuronal responses to various conditions, quantitative reproduction as well as prediction for novel experiments appears to be difﬁcult to realize [7]. [sent-18, score-0.149]

12 The difﬁculty is due to the complexity of the model accompanied with a large number of undetermined free parameters. [sent-19, score-0.034]

13 Even if a true model of a particular neuron is included in the family of models, it is practically difﬁcult to explore the true parameters in the high-dimensional space of parameters that dominate the nonlinear dynamics. [sent-20, score-0.405]

14 Recently it was suggested by Kistler et al [8, 9] to extend the leaky integrate-and-ﬁre model so that real membrane dynamics of any neuron can be adopted. [sent-21, score-0.868]

15 The so called “spike response model” has been successful in not only reproducing the data but also in predicting the spike timing for a novel input current [8, 9, 10, 11]. [sent-22, score-0.703]

16 The fairly precise prediction achieved by such a simple model indicates that the spike occurrence is determined principally by the subthreshold dynamics. [sent-24, score-1.088]

17 In other words, the highly nonlinear dynamics of a neuron can be decomposed into two simple, predictable processes: a relatively simple subthreshold dynamics, and the dynamics of an action potential of a nearly ﬁxed shape (Fig. [sent-25, score-0.936]

18 action potential (nearly fixed) V subthreshold (predictable) dV/dt Figure 1: The highly nonlinear dynamics of a neuron is decomposed into two simple, predictable processes. [sent-27, score-0.83]

19 In this paper, we propose a framework of improving the prediction of spike times by paying close attention to the transfer between the two predictable processes mentioned above. [sent-28, score-0.7]

20 It is assumed in the original spike response model that a spike occurs if the membrane potential exceeds a certain threshold [9]. [sent-29, score-1.687]

21 We revised this rule to maximally utilize the information of a higher-dimensional state space, consisting of not only the instantaneous membrane potential, but also its time derivative(s). [sent-30, score-0.451]

22 Such a subthreshold state can provide cues for the occurrence of a spike, but with a certain time difference. [sent-31, score-0.569]

23 For the purpose of exploring the optimal time shift, we propose a method of maximizing the mutual information between the subthreshold state and the occurrence of a spike. [sent-32, score-0.591]

24 By employing the linear ﬁlter model [12] and the spike response model [9] for mimicking the subthreshold voltage response of a neuron, we examine how much the present framework may improve the prediction for simulation data of the fast-spiking model [13]. [sent-33, score-1.202]

25 2 Methods The response of a neuron is precisely reproduced when presented with identical ﬂuctuating input currents [14]. [sent-34, score-0.553]

26 This implies that the neuronal membrane potential V (t) is determined by the past input current {I(t)}, or V (t) = F ({I(t)}), (1) where F ({I(t)}) represents a functional of a time-dependent current I(t). [sent-35, score-0.491]

27 A rapid swing in the polarity of the membrane potential is called a “spike. [sent-36, score-0.401]

28 ” The occurrence of a spike could be deﬁned practically by measuring the membrane potential V (t) exceeding a certain threshold, V (t) > Vth . [sent-37, score-1.203]

29 (2) The time of each spike could be deﬁned either as the ﬁrst time the threshold is exceeded, or as the peak of the action potential that follows the crossing. [sent-38, score-0.791]

30 Kistler et al [8] and Jolivet et al [10, 11] proposed a method of mimicking the membrane dynamics of a target neuron with the simple spike response model in which an input current is linearly integrated. [sent-39, score-1.697]

31 The leaky integrate-and-ﬁre model can be regarded as an example of the spike response model [9]; the differential equation can be rewritten as an integral equation in which the membrane potential is given as the integral of the past input current with an exponentially decaying kernel. [sent-40, score-1.201]

32 The spike response model is an extension of the leaky integrate-and-ﬁre model, where the integrating kernel is adaptively determined by the data, and the after hyperpolarizing potential is added subsequently to every spike. [sent-41, score-0.85]

33 It is also possible to further include terms that reduce the responsiveness and increase the threshold after an action potential takes place. [sent-42, score-0.215]

34 Even in the learning stage, no model is able to perfectly reproduce the output V (t) of a target neuron for a given input I(t). [sent-43, score-0.423]

35 We will denote the output of the model (in the lower case) as v(t) = fk ({I(t)}), (3) where k represents a set of model parameters. [sent-44, score-0.068]

36 The model parameters are learned by mimicking sample input-output data. [sent-45, score-0.096]

37 1 State space method As the output of the model v(t) is not identical to the true membrane potential of the target neuron V (t), a spike occurrence cannot be determined accurately by simply applying the same threshold rule Eq. [sent-48, score-1.649]

38 In this paper, we suggest revising the spike generation rule so that a spike occurrence is best predicted from the model potential v(t). [sent-50, score-1.497]

39 Suppose that we have adjusted the parameters of fk ({I(t)}) so that the output of the model {v(t)} best approximates the membrane potential {V (t)} of a target neuron for a given set of currents {I(t)}. [sent-51, score-0.986]

40 If the sample data set {I(t), V (t)} employed in learning is large enough, the spike occurrence can be predicted by estimating an empirical probability of a spike being generated at the time t, given a time-dependent orbit of an estimated output, {v(t)}, as pspike (t|{v(t)}). [sent-52, score-1.677]

41 (5) In a practical experiment, however, the amount of collectable data is insufﬁcient for estimating the spiking probability with respect to any orbit of v(t). [sent-53, score-0.275]

42 In place of such exhaustive examination, we suggest utilizing the state space information such as the time derivatives of the model potential at a certain time. [sent-54, score-0.345]

43 The spike occurrence at time t could be predicted from the m-dimensional state space information ⃗ ≡ (v, v ′ , · · · , v (m−1) ), as observed at a time s before t, as v pspike (t|⃗t−s ), v (6) where ⃗t−s ≡ (v(t − s), v ′ (t − s), · · · , v (m−1) (t − s)). [sent-55, score-1.11]

44 2 Determination of the optimal time shift The time shift s introduced in the spike time prediction, Eq. [sent-57, score-0.68]

45 We propose optimizing the time shift s by maximizing the mutual information between the state space information ⃗t−s and the presence or absence of a spike at a time interval v (t − δt/2, t + δt/2], which is denoted as zt = 1 or 0. [sent-59, score-1.014]

46 The mutual information [15] is given as M I(zt ; ⃗t−s ) = M I(⃗t−s ; zt ) = H(⃗t−s ) − H(⃗t−s |zt ), v v v v where (7) ∫ H(⃗t−s ) v H(⃗t−s |zt ) v = − d⃗t−s p(⃗t−s ) log p(⃗t−s ), v v v ∫ ∑ = − d⃗t−s p(⃗t−s |zt )p(zt ) log p(⃗t−s |zt ). [sent-60, score-0.254]

47 v v v (8) (9) zt ∈{0,1} Here, p(⃗t−s |zt ) is the probability, given the presence or absence of a spike at time t ∈ (t−δt/2, t+ v δt/2], of the state being ⃗t−s , a time s before the spike. [sent-61, score-0.888]

48 v With the time difference s optimized, we then obtain the empirical probability of the spike occurrence at the time t, given the state space information at the time t − s, using the Bayes theorem, pspike (t|⃗t−s ) ∝ p(zt = 1|⃗t−s ) = v v 3 p(⃗t−s |zt )p(zt ) v . [sent-62, score-1.092]

49 p(⃗t−s ) v (10) Results We evaluated our state space method of predicting spike times by applying it to target data obtained for a fast-spiking neuron model proposed by Erisir et al [13] (see Appendix). [sent-63, score-1.217]

50 In this virtual experiment, two ﬂuctuating currents characterized by the same mean and standard deviation are injected to the (model) fast-spiking neuron to obtain two sets of input-output data {I(t), V (t)}. [sent-64, score-0.49]

51 A predictive model was trained using one sample data set, and then its predictive performance for the other sample data was evaluated. [sent-65, score-0.126]

52 Each input current is generated by the Ornstein-Uhlenbeck process. [sent-66, score-0.028]

53 We have tested two kinds of ﬂuctuating currents characterized with different means and standard deviations: (Currents I) the mean µ = 1. [sent-67, score-0.183]

54 0 [µA] and the time scale of the ﬂuctuation τ = 2 [msec]; (Currents II) the mean µ = 0. [sent-69, score-0.026]

55 0 [µA] and the time scale of the ﬂuctuation τ = 2 [msec]. [sent-71, score-0.026]

56 In this study we adopted the linear ﬁlter model and the spike response model as prediction models. [sent-73, score-0.794]

57 B: The mutual information between the estimated potential and the occurrence of a spike. [sent-78, score-0.411]

58 We brieﬂy describe here the results for the linear ﬁlter model [12], ∫ ∞ v(t) = K(t′ )I(t − t′ ) dt′ + v0 . [sent-79, score-0.034]

59 (11) 0 The model parameters k consist of the shape of the kernel K(t) and the constant v0 . [sent-80, score-0.06]

60 In learning the target sample data {I(t), V (t)}, these parameters are adjusted to minimize the integrated square error, Eq. [sent-81, score-0.106]

61 Figure 2A depicts the shape of the kernel K(t) estimated from the target sample data {I(t), V (t)} obtained from the virtual experiment of the fast-spiking neuron model. [sent-83, score-0.451]

62 Based on the voltage estimation v(t) with respect to sample data, we compute the empirical probabilities, p(⃗t−s ), p(⃗t−s |zt ) and p(zt ) for two-dimensional state space information ⃗t−s ≡ v v v (v(t − s), v ′ (t − s)). [sent-84, score-0.195]

63 In computing empirical probabilities, we quantized the two-dimensional phase space ⃗ ≡ (v, v ′ ), and the time. [sent-85, score-0.042]

64 In the discretized time, we counted the occurrence of a spike, v zt = 1, for the bins in which the true membrane potential V (t) exceeds a reasonable threshold Vth . [sent-86, score-0.937]

65 With a sufﬁciently small time step (we adopted δt = 0. [sent-87, score-0.06]

66 1 [msec]), a single spike is transformed into a succession of spike occurrences zt = 1. [sent-88, score-1.257]

67 2B whose maximum position of s ≈ 2 [msec] determines the optimal time shift. [sent-91, score-0.026]

68 The spike is predicted if the estimated probability pspike (t|⃗t−s ) of Eq. [sent-92, score-0.71]

69 Though it would be more efﬁcient to use the systematic method suggested by Paninski et al [16], we determined the threshold value empirically so that the coincidence factor Γ described in the following is maximized. [sent-94, score-0.335]

70 Figure 3 compares a naive thresholding method and our state space method, in reference to the original spike times. [sent-95, score-0.825]

71 It is observed from this ﬁgure that the prediction of the state space method is more accurate and robust than that of thresholding method. [sent-96, score-0.36]

72 50 V(t) 0 50 v(t) 0 pspike(t) 3000 3500 t (ms) Figure 3: Comparison of the spike time predictions. [sent-97, score-0.55]

73 Figure 4 depicts an orbit in the state space of (V, V ′ ) of a target neuron for an instance of the spike generation, and the orbit of the predictive model in the state space of (v, v ′ ) that mimics it. [sent-102, score-1.69]

74 The predictive model can mimic the target orbit in the subthreshold region, but fails to catch the spiking orbit in the suprathreshold region. [sent-103, score-0.804]

75 The spike occurrence is predicted by estimating the conditional probability, Eq. [sent-104, score-0.794]

76 (10), given the state (v, v ′ ) of the predictive model. [sent-105, score-0.149]

77 The contour lines for higher probabilities of spiking resemble an ad hoc “dynamic spike threshold” introduced by Azouz and Gray [17]. [sent-107, score-0.698]

78 Namely, v drops with dv/dt along the contour lines. [sent-108, score-0.065]

79 Contrastingly, the contour lines for lower probabilities of spiking are inversely curved: v increases with dv/dt along the contour lines. [sent-109, score-0.239]

80 In the present framework, the state space information corresponding to the relatively low probability of spiking is effectively used for predicting spike times. [sent-110, score-0.8]

81 ∆ is chosen as 2 [msec] in accordance with Jolivet et al [10]. [sent-112, score-0.138]

82 Table 1: The coincidence factors evaluated for two methods of prediction based on the linear ﬁlter model. [sent-113, score-0.192]

83 666 B C v V c 14 14 b A c 12 12 10 10 8 b 8 V 120 80 40 a a 6 0 -500 0 500 dV/dt 6 -2 0 dV/dt 2 -2 0 dv/dt 2 Figure 4: A: An orbit in the state space of (V, V ′ ) of a target neuron for an instance of the spike generation (from 3240 to 3270 [msec] of Fig. [sent-118, score-1.217]

84 C: The orbit in the state space of (v, v ′ ) of the predictive model that mimics the target neuron. [sent-121, score-0.54]

85 Contours represent the probability of spike occurrence computed with the Bayes formula, Eq. [sent-122, score-0.75]

86 The dashed lines represent the threshold adopted in the naive thresholding method (Fig 3 Middle). [sent-124, score-0.283]

87 Table 2: The coincidence factors evaluated for two methods of prediction based on the spike response model. [sent-126, score-0.796]

88 842 The coincidence factors evaluated for a simple thresholding method and the state space method based on the linear ﬁlter model are summarized in Table 1, and those based on the spike response model are summarized in Table 2. [sent-131, score-1.048]

89 It is observed that the prediction is signiﬁcantly improved by our state space method. [sent-132, score-0.233]

90 It should be noted, however, that a model with the same set of parameters does not perform well over a range of inputs generated with different mean and variance: The model parameterized with the Currents I does not effectively predict the spikes of the neuron for the Current II, and vice versa. [sent-133, score-0.379]

91 Nevertheless, our state-space method exhibits the better prediction than the naive thresholding strategy, if the statistics of the different inputs are relatively similar. [sent-134, score-0.244]

92 4 Summary We proposed a method of evaluating the probability of the spike occurrence by observing the state space of the membrane potential and its time derivative(s) in advance of the possible spike time. [sent-135, score-1.888]

93 It is found that the prediction is signiﬁcantly improved by the state space method compared to the prediction obtained by simply thresholding an instantaneous value of the estimated potential. [sent-136, score-0.509]

94 It is interesting to apply our method to biological data and categorize neurons based on their spiking mechanisms. [sent-137, score-0.084]

95 The state space method developed here is a rather general framework that may be applicable to any nonlinear phenomena composed of locally predictable dynamics. [sent-138, score-0.309]

96 The generalization of linear ﬁlter analysis developed here has a certain similarity to the Linear-Nonlinear-Poisson (LNP) model [18, 19]. [sent-139, score-0.057]

97 It would be interesting to generalize the present method of analysis to a wider range of phenomena such as the analysis of the coding of visual system [19, 20]. [sent-140, score-0.038]

98 Appendix: Fast-spiking neuron model The fast-spiking neuron model proposed by Erisir et al [13] was used in this contribution as a (virtual) target experiment. [sent-142, score-0.797]

99 The details of the model were adjusted to Jolivet et al [10] to allow the direct comparison of the performances. [sent-143, score-0.211]

100 (2006) Integrate-and-Fire models with u adaptation are good enough: predicting spike times under random current injection. [sent-230, score-0.571]

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