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115 nips-2010-Identifying Dendritic Processing

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Author: Aurel A. Lazar, Yevgeniy Slutskiy

Abstract: In system identiﬁcation both the input and the output of a system are available to an observer and an algorithm is sought to identify parameters of a hypothesized model of that system. Here we present a novel formal methodology for identifying dendritic processing in a neural circuit consisting of a linear dendritic processing ﬁlter in cascade with a spiking neuron model. The input to the circuit is an analog signal that belongs to the space of bandlimited functions. The output is a time sequence associated with the spike train. We derive an algorithm for identiﬁcation of the dendritic processing ﬁlter and reconstruct its kernel with arbitrary precision. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract In system identiﬁcation both the input and the output of a system are available to an observer and an algorithm is sought to identify parameters of a hypothesized model of that system. [sent-6, score-0.097]

2 Here we present a novel formal methodology for identifying dendritic processing in a neural circuit consisting of a linear dendritic processing ﬁlter in cascade with a spiking neuron model. [sent-7, score-1.417]

3 The input to the circuit is an analog signal that belongs to the space of bandlimited functions. [sent-8, score-0.403]

4 The output is a time sequence associated with the spike train. [sent-9, score-0.175]

5 We derive an algorithm for identiﬁcation of the dendritic processing ﬁlter and reconstruct its kernel with arbitrary precision. [sent-10, score-0.351]

6 1 Introduction The nature of encoding and processing of sensory information in the visual, auditory and olfactory systems has been extensively investigated in the systems neuroscience literature. [sent-11, score-0.164]

7 Many phenomenological [1, 2, 3] as well as mechanistic [4, 5, 6] models have been proposed to characterize and clarify the representation of sensory information on the level of single neurons. [sent-12, score-0.089]

8 Here we investigate a class of phenomenological neural circuit models in which the time-domain linear processing takes place in the dendritic tree and the resulting aggregate dendritic current is encoded in the spike domain by a spiking neuron. [sent-13, score-1.295]

9 While the LNP model also includes a linear processing stage, it describes spike generation using an inhomogeneous Poisson process. [sent-15, score-0.209]

10 In contrast, the [Filter]-[Spiking Neuron] model incorporates the temporal dynamics of spike generation and allows one to consider more biologically-plausible spike generators. [sent-16, score-0.37]

11 We perform identiﬁcation of dendritic processing in the [Filter]-[Spiking Neuron] model assuming that input signals belong to the space of bandlimited functions, a class of functions that closely model natural stimuli in sensory systems. [sent-17, score-0.521]

12 Under this assumption, we show that the identiﬁcation of dendritic processing in the above neural circuit becomes mathematically tractable. [sent-18, score-0.63]

13 Using simulated data, we demonstrate that under certain conditions it is possible to identify the impulse response of the dendritic processing ﬁlter with arbitrary precision. [sent-19, score-0.469]

14 The phenomenological neural circuit model and the identiﬁcation problem are formally stated in section 2. [sent-22, score-0.344]

15 The Neural Identiﬁcation Machine and its realization as an algorithm for identifying dendritic processing is extensively discussed in section 3. [sent-23, score-0.376]

16 1 2 Problem Statement In what follows we assume that the dendritic processing is linear [11] and any nonlinear effects arise as a result of the spike generation mechanism [12]. [sent-27, score-0.516]

17 We use linear BIBO-stable ﬁlters (not necessarily causal) to describe the computation performed by the dendritic tree. [sent-28, score-0.307]

18 Furthermore, a spiking neuron model (as opposed to a rate model) is used to model the generation of action potentials or spikes. [sent-29, score-0.394]

19 We investigate a general neural circuit comprised of a ﬁlter in cascade with a spiking neuron model (Fig. [sent-30, score-0.709]

20 This circuit is an instance of a Time Encoding Machine (TEM), a nonlinear asynchronous circuit that encodes analog signals in the time domain [13, 14]. [sent-32, score-0.586]

21 Examples of spiking neuron models considered in this paper include the ideal IAF neuron, the leaky IAF neuron and the threshold-and-feedback (TAF) neuron [15]. [sent-33, score-0.964]

22 However, the methodology developed below can be extended to many other spiking neuron models as well. [sent-34, score-0.38]

23 We break down the full identiﬁcation of this circuit into two problems: (i) identiﬁcation of linear operations in the dendritic tree and (ii) identiﬁcation of spike generator parameters. [sent-35, score-0.757]

24 First, we consider problem (i) and assume that parameters of the spike generator can be obtained through biophysical experiments. [sent-36, score-0.2]

25 We consider a speciﬁc example of a neural circuit in Fig. [sent-38, score-0.302]

26 Dendritic Processing u(t) Linear F ilter Dendritic Processing Spike Generation v(t) Spiking N euron u(t) h(t) Spike Generation: Ideal IAF Neuron v(t) 1 C + (tk )k∈Z b (a) δ (tk )k∈Z voltage reset to 0 (b) Figure 1: Problem setup. [sent-40, score-0.081]

27 (a) The dendritic processing is described by a linear ﬁlter and spikes are produced by a (nonlinear) spiking neuron model. [sent-41, score-0.733]

28 (b) An example of a neural circuit in (a) is a linear ﬁlter in cascade with the ideal IAF neuron. [sent-42, score-0.477]

29 An input signal u is ﬁrst passed through a ﬁlter with an impulse response h. [sent-43, score-0.183]

30 The output of the ﬁlter v(t) = (u ∗ h)(t), t ∈ R, is then encoded into a time sequence (tk )k∈Z by the ideal IAF neuron. [sent-44, score-0.172]

31 3 Neuron Identiﬁcation Machines A Neuron Identiﬁcation Machine (NIM) is the realization of an algorithm for the identiﬁcation of the dendritic processing ﬁlter in cascade with a spiking neuron model. [sent-45, score-0.753]

32 First, we introduce several deﬁnitions needed to formally address the problem of identifying dendritic processing. [sent-46, score-0.337]

33 We derive an algorithm for a perfect identiﬁcation of the impulse response of the ﬁlter and provide conditions for the identiﬁcation with arbitrary precision. [sent-48, score-0.139]

34 1 Preliminaries We model signals u = u(t), t ∈ R, at the input to a neural circuit as elements of the Paley-Wiener space Ξ = u ∈ L2 (R) supp (Fu) ⊆ [−Ω, Ω] , i. [sent-51, score-0.437]

35 Furthermore, we assume that the dendritic processing ﬁlters h = h(t), t ∈ R, are linear, BIBO-stable and have a ﬁnite temporal support, i. [sent-54, score-0.358]

36 A signal u ∈ Ξ at the input to a neural circuit together with the resulting output T = (tk )k∈Z of that circuit is called an input/output (I/O) pair and is denoted by (u, T). [sent-58, score-0.662]

37 Two neural circuits are said to be Ξ-I/O-equivalent if their respective I/O pairs are identical for all u ∈ Ξ. [sent-60, score-0.056]

38 Signals {ui }N are said to be linearly independent if there do not exist real numbers i=1 N {αi }N , not all zero, and real numbers {βi }N such that i=1 αi ui (t + βi ) = 0. [sent-65, score-0.066]

39 2 NIM for the [Filter]-[Ideal IAF] Neural Circuit An example of a model circuit in Fig. [sent-67, score-0.274]

40 1(a) is the [Filter]-[Ideal IAF] circuit shown in Fig. [sent-68, score-0.274]

41 In this circuit, an input signal u ∈ Ξ is passed through a ﬁlter with an impulse response (kernel) h ∈ H and then encoded by an ideal IAF neuron with a bias b ∈ R+ , a capacitance C ∈ R+ and a threshold δ ∈ R+ . [sent-70, score-0.549]

42 The output of the circuit is a sequence of spike times (tk )k∈Z that is available to an observer. [sent-71, score-0.449]

43 This neural circuit is an instance of a TEM and its operation can be described by a set of equations (formally known as the t-transform [13]): tk+1 tk (u ∗ h)(s)ds = qk , k ∈ Z, (1) where qk Cδ−b(tk+1 −tk ). [sent-72, score-0.903]

44 Intuitively, at every spike time tk+1 the ideal IAF neuron is providing a measurement qk of the signal v(t) = (u ∗ h)(t) on the interval t ∈ [tk , tk+1 ]. [sent-73, score-0.622]

45 The left-hand side of the t-transform in (1) can be written as a bounded linear functional Lk : Ξ → R with Lk (Ph) = φk , Ph , where φk (t) = 1[tk , tk+1 ] ∗ u (t) and u = ˜ ˜ u(−t), t ∈ R, denotes the involution of u. [sent-75, score-0.056]

46 tk tk+1 (u∗h)(s)ds tk = Now since Ph is bounded, the expression on the right-hand side of the equality is a bounded linear functional Lk : Ξ → R with tk+1 Lk (Ph) = tk (u ∗ Ph)(s)ds = φk , Ph , (2) where φk ∈ Ξ and the last equality follows from the Riesz representation theorem [16]. [sent-77, score-1.263]

47 By the reproducing property of the kernel [17], we have φk (t) = φk , Kt = Lk (Kt ). [sent-79, score-0.044]

48 Letting u = u(−t) denote the involution of u and using (2), we obtain ˜ φk (t) = 1[tk , tk+1 ] ∗ u, Kt = 1[tk , tk+1 ] ∗ u (t). [sent-80, score-0.032]

49 To that end, we note that an observer can typically record both the input u = u(t), t ∈ R and the output T = (tk )k∈Z of a neural circuit. [sent-83, score-0.104]

50 Since (qk )k∈Z can be evaluated from (tk )k∈Z using the deﬁnition of qk in (1), the problem is reduced to identifying Ph from an I/O pair (u, T). [sent-84, score-0.124]

51 Then given an I/O pair (u, T) of the [Filter]-[Ideal IAF] neural circuit, Ph can be perfectly identiﬁed as (Ph)(t) = ck ψk (t), k∈Z where ψk (t) = g(t − tk ), t ∈ R. [sent-87, score-0.532]

52 Furthermore, c = G+ q with G+ denoting the Moore-Penrose t pseudoinverse of G, [G]lk = tll+1 u(s − tk )ds for all k, l ∈ Z, and [q]l = Cδ − b(tl+1 − tl ). [sent-88, score-0.559]

53 Proof: By appropriately bounding the input signal u, the spike density (the average number of spikes over arbitrarily long time intervals) of an ideal IAF neuron is given by D = b/(Cδ) [14]. [sent-89, score-0.632]

54 Therefore, for D > Ω/π the set of the representation functions (ψk )k∈Z , ψk (t) = g(t − tk ), is a frame in Ξ [18] and (Ph)(t) = k∈Z ck ψk (t). [sent-90, score-0.481]

55 To ﬁnd the coefﬁcients ck we note from (2) that ql = φl , Ph = ck φl , ψk = k∈Z [G]lk ck , (3) k∈Z t where [G]lk = φl , ψk = 1[tl , tl+1 ] ∗ u, g( · − tk ) = tll+1 u(s − tk )ds. [sent-91, score-1.09]

56 Writing (3) in matrix ˜ form, we obtain q = Gc with [q]l = ql and [c]k = ck . [sent-92, score-0.128]

57 Finally, the coefﬁcients ck , k ∈ Z, can be computed as c = G+ q. [sent-93, score-0.068]

58 Thus, perfect identiﬁcation of the projection of h onto Ξ can be achieved for a ﬁnite average spike rate. [sent-96, score-0.214]

59 Ideally, we would like to identify the kernel h ∈ H of the ﬁlter in cascade with the ideal IAF neuron. [sent-98, score-0.219]

60 Nevertheless, it is easy to show that (Ph)(t) approximates h(t) arbitrarily closely on t ∈ [T1 , T2 ], provided that the bandwidth Ω of u is sufﬁciently large. [sent-102, score-0.07]

61 , if there is no processing on the (arbitrary) t t input signal u(t), then ql = tll+1 (u ∗ h)(s)ds = tll+1 u(s)ds, l ∈ Z. [sent-106, score-0.144]

62 Furthermore, tl+1 tl tl+1 (u ∗ Ph)(s)ds = tl tl+1 (u ∗ h)(s)ds = tl+1 (u ∗ g)(s)ds, u(s)ds = tl tl l ∈ Z. [sent-107, score-0.584]

63 In other words, if h(t) = δ(t), then we identify Pδ(t) = sin(Ωt)/(πt), the projection of δ(t) onto Ξ. [sent-109, score-0.064]

64 2, we obtain W = [τ1 − τ + T, τ2 − τ + T ] ⊃ [T1 , T2 ] and the set of spike times (tk − τ + T )k: tk ∈W . [sent-115, score-0.565]

65 From Corollary 1 we see that if the [Filter]-[Ideal IAF] neural circuit is producing spikes with a spike density above the Nyquist rate, then we can use a set of spike times (tk )k: tk ∈W from a single temporal window W to identify (Ph)(t) to an arbitrary precision on [T1 , T2 ]. [sent-118, score-1.117]

66 Since the spike density is above the Nyquist rate, we could have also used a canonical time decoding machine (TDM) [13] to ﬁrst perfectly recover the ﬁlter output v(t) and then employ one of the widely available LTI system techniques to estimate (Ph)(t). [sent-120, score-0.198]

67 However, the problem becomes much more difﬁcult if the spike density is below the Nyquist rate. [sent-121, score-0.152]

68 (a) Top: example of a causal impulse response h(t) with supp(h) = [T1 , T2 ], T1 = 0. [sent-123, score-0.153]

69 (b) Top: an input signal u(t) with supp(Fu) = [−Ω, Ω]. [sent-127, score-0.063]

70 Middle: only red spikes from a temporal window W = (τ1 , τ2 ) are used to ˆ ˆ construct h(t). [sent-128, score-0.077]

71 Bottom: Ph is approximated by h(t) on t ∈ [T1 , T2 ] using spike times (tk − τ + T )k:tk ∈W . [sent-129, score-0.173]

72 (The Neuron Identiﬁcation Machine) Let {ui | supp(Fui ) = [−Ω, Ω] }N be a coli=1 lection of N linearly independent and bounded stimuli at the input to a [Filter]-[Ideal IAF] neural circuit with a dendritic processing ﬁlter h ∈ H. [sent-131, score-0.712]

73 Furthermore, let Ti = (ti )k∈Z denote the output of k N b the neural circuit in response to the bounded input signal ui . [sent-132, score-0.514]

74 h(t) is the input to a population of N [Filter]-[Ideal IAF] neural circuits. [sent-136, score-0.058]

75 The spikes (ti )k∈Z at the output of each neural circuit represent k i i distinct measurements qk = φi , Ph of (Ph)(t). [sent-137, score-0.466]

76 Thus we can think of the qk ’s as projections k 1 1 1 N N N of Ph onto (φ1 , φ2 , . [sent-138, score-0.117]

77 Since the ﬁlters are linearly independent N b [14], it follows that, if {ui }N are appropriately bounded and j=1 Cδ > Ω or equivalently if the i=1 π j j Ω number of neurons N > ΩCδ = πD , the set of functions { (ψk )k∈Z }N with ψk (t) = g(t − tj ), is j=1 k πb a frame for Ξ [14], [18]. [sent-151, score-0.128]

78 k (Ph)(t) = (4) j=1 k∈Z To ﬁnd the coefﬁcients ck , we take the inner product of (4) with φ1 (t), φ2 (t), . [sent-153, score-0.068]

79 Letting [Gij ]lk = i ql = Gi1 k∈Z c1 lk k Gi2 + N cN φi , ψk k l k∈Z ≡ i ql , , we obtain c2 lk k k∈Z + ··· + GiN cN , lk k (5) k∈Z for i = 1, . [sent-160, score-0.618]

80 , qN ]T with [qi ]l = Cδ − b(ti − ti ), [Gij ]lk = l+1 l ti l+1 ti l ui (s − tj )ds and c = [c1 , c2 , . [sent-167, score-0.562]

81 Finally, k to ﬁnd the coefﬁcients ck , k ∈ Z, we compute c = G+ q. [sent-171, score-0.068]

82 Furthermore, let τ i = (τ1 + τ2 )/2, i i T = (T1 + T2 )/2 and let (tk )k∈Z denote those spikes of the I/O pair (u, T) that belong to W . [sent-185, score-0.047]

83 Then Ph can be approximated arbitrarily closely on [T1 , T2 ] by N ˆ h(t) = j cj ψk (t), k j=1 k: tk ∈W j j where ψk (t) = g(t − (tj − τ j + T )), c = G+ q with [Gij ]lk = k ti l+1 ti l u(s − (tj − τ j + T ))ds, k q = [q1 , q2 , . [sent-186, score-0.788]

84 , qN ]T , [qi ]l = Cδ − b(ti − ti ) for all k, l ∈ Z, provided that the number of l+1 l non-overlapping windows N is sufﬁciently large. [sent-189, score-0.142]

85 i i Proof: The input signal u restricted, respectively, to the collection of intervals W i τ1 , τ2 i N plays the same role here as the test stimuli {u }i=1 in Corollary 2. [sent-190, score-0.091]

86 The methodology presented in Theorem 2 can easily be applied to other spiking neuron models. [sent-199, score-0.38]

87 For example, for the leaky IAF neuron, we have ti − ti l+1 [q ]l = Cδ − bRC 1 − exp l RC i ti l+1 ij , [G ]lk = ti l ui s − tj exp k s − ti l+1 ds. [sent-200, score-0.892]

88 RC Similarly, for a threshold-and-feedback (TAF) neuron [15] with a bias b ∈ R+ , a threshold δ ∈ R+ , and a causal feedback ﬁlter with an impulse response f (t), t ∈ R, we obtain [qi ]l = δ − b + 3. [sent-201, score-0.37]

89 5 Conclusion Previous work in system identiﬁcation of neural circuits (see [20] and references therein) calls for parameter identiﬁcation using white noise input stimuli. [sent-203, score-0.086]

90 In our work, we presented the methodology for identifying dendritic processing in simple [Filter][Spiking Neuron] models from a single input stimulus. [sent-208, score-0.41]

91 The discussed spiking neurons include the ideal IAF neuron, the leaky IAF neuron and the threshold-and-ﬁre neuron. [sent-209, score-0.564]

92 However, the methods presented in this paper are applicable to many other spiking neuron models as well. [sent-210, score-0.358]

93 The algorithm of the Neuron Identiﬁcation Machine is based on the natural assumption that the dendritic processing ﬁlter has a ﬁnite temporal support. [sent-211, score-0.358]

94 Therefore, its action on the input stimulus can be observed in non-overlapping temporal windows. [sent-212, score-0.06]

95 The ﬁlter is recovered with arbitrary precision from an input/output pair of a neural circuit, where the input is a single signal assumed to be bandlimited. [sent-213, score-0.091]

96 Finally, the work presented here will be extended to spiking neurons with random parameters. [sent-216, score-0.175]

97 Computational model of the cAMPmediated sensory response and calcium-dependent adaptation in vertebrate olfactory receptor neurons. [sent-241, score-0.131]

98 Quantitative characterization procedure for auditory neurons based on the spectra-temporal receptive ﬁeld. [sent-263, score-0.061]

99 Perfect recovery and sensitivity analysis of time encoded bandlimited o signals. [sent-280, score-0.071]

100 Complete functional characterization of sensory neurons by system identiﬁcation. [sent-312, score-0.081]

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