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200 nips-2011-On the Analysis of Multi-Channel Neural Spike Data

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Author: Bo Chen, David E. Carlson, Lawrence Carin

Abstract: Nonparametric Bayesian methods are developed for analysis of multi-channel spike-train data, with the feature learning and spike sorting performed jointly. The feature learning and sorting are performed simultaneously across all channels. Dictionary learning is implemented via the beta-Bernoulli process, with spike sorting performed via the dynamic hierarchical Dirichlet process (dHDP), with these two models coupled. The dHDP is augmented to eliminate refractoryperiod violations, it allows the “appearance” and “disappearance” of neurons over time, and it models smooth variation in the spike statistics. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Nonparametric Bayesian methods are developed for analysis of multi-channel spike-train data, with the feature learning and spike sorting performed jointly. [sent-3, score-0.874]

2 The feature learning and sorting are performed simultaneously across all channels. [sent-4, score-0.472]

3 Dictionary learning is implemented via the beta-Bernoulli process, with spike sorting performed via the dynamic hierarchical Dirichlet process (dHDP), with these two models coupled. [sent-5, score-0.958]

4 The dHDP is augmented to eliminate refractoryperiod violations, it allows the “appearance” and “disappearance” of neurons over time, and it models smooth variation in the spike statistics. [sent-6, score-0.552]

5 1 Introduction The analysis of action potentials (“spikes”) from neural-recording devices is a problem of longstanding interest (see [21, 1, 16, 22, 8, 4, 6] and the references therein). [sent-7, score-0.045]

6 In such research one is typically interested in clustering (sorting) the spikes, with the goal of linking a given cluster to a particular neuron. [sent-8, score-0.146]

7 Such technology is of interest for brain-machine interfaces and for gaining insight into the properties of neural circuits [14]. [sent-9, score-0.073]

8 In such research one typically (i) ﬁlters the raw sensor readings, (ii) performs thresholding to “detect” the spikes, (iii) maps each detected spike to a feature vector, and (iv) then clusters the feature vectors [12]. [sent-10, score-0.78]

9 Principal component analysis (PCA) is a popular choice [12] for feature mapping. [sent-11, score-0.049]

10 Many of the early methods for spike sorting were based on classical clustering techniques [12] (e. [sent-14, score-0.844]

11 However, [5] assumed that the spike features were provided via PCA in the ﬁrst two or three principal components (PCs). [sent-18, score-0.559]

12 In [6] feature learning and spike sorting were performed jointly via a mixture of factor analyzers (MFA) formulation. [sent-19, score-0.987]

13 There has been an increasing interest in developing neural devices with C > 1 recording channels, each of which produces a separate electrical recording of neural activity. [sent-21, score-0.319]

14 (a) Ground truth; (b) K-means clustering on the ﬁrst 2 principal components; (c) GMM clustering with the ﬁrst 2 principal components; (d) proposed method. [sent-24, score-0.234]

15 We label using arrows examples K-means and the GMM miss, and that the proposed method properly sort. [sent-25, score-0.034]

16 has been performed on a single channel, or when multiple channels are present each is typically analyzed in isolation. [sent-26, score-0.226]

17 In [5] C = 4 channels were considered, but it was assumed that a spike occurred at the same time (or nearly same time) across all channels, and the features from the four channels were concatenated, effectively reducing this again to a single-channel analysis. [sent-27, score-0.863]

18 When C 1, the assumption that a given neuron is observed simultaneously on all channels is typically inappropriate, and in fact the diversity of neuron sensing across the device is desired, to enhance functionality [18]. [sent-28, score-0.593]

19 This paper addresses the multi-channel neural-recording problem, under conditions for which concatenation may be inappropriate; the proposed model generalizes the DP formulation of [5], with a hierarchical DP (HDP) formulation [20]. [sent-29, score-0.043]

20 In this formulation statistical strength is shared across the channels, without assuming that a given neuron is simultaneously viewed across all channels. [sent-30, score-0.285]

21 Further, the model generalizes the HDP, via a dynamic HDP (dHDP) [17] to allow the “appearance”/“disappearance” of neurons, while also allowing smooth changes in the statistics of the neurons. [sent-31, score-0.09]

22 Further, we explicitly account for refractory times, as in [5]. [sent-32, score-0.043]

23 We also perform joint feature learning and clustering, using a mixture of factor analyzers construction as in [6], but we do so in a fully Bayesian, multi-channel setting (additionally, [6] did not account for time-varying statistics). [sent-33, score-0.166]

24 The learned factor loadings are found to be similar to wavelets, but they are matched to the properties of neuron spikes; this is in contrast to previous feature extraction on spikes [11] based on orthogonal wavelets, that are not necessarily matched to neuron properties. [sent-34, score-0.785]

25 To give a preview of the results, providing a sense of the importance of feature learning (relative to mapping data into PCA features learned ofﬂine), in Figure 1 we show a comparison of clustering results on the ﬁrst channel of d533101 data from hc-1 [7]. [sent-35, score-0.301]

26 For all cases in Figure 1 the data are depicted in the ﬁrst two PCs for visualization, but the proposed method in (d) learns the number of features and their composition, while simultaneously performing clustering. [sent-36, score-0.029]

27 The results in (b) and (c) correspond respectively to widely employed K-means and GMM analysis, based on using two PCs (in these cases the analysis are employed in PCA space, as have been many more-advanced approaches [5]). [sent-37, score-0.078]

28 From Figures 1 (b) and (c), we observe that both K-means and GMM work well, but due to the constrained feature space they incorrectly classify some spikes (marked by arrows). [sent-38, score-0.334]

29 However, the proposed model, shown in Figure 1(d), which incorporates dictionary learning with spike sorting, infers an appropriate feature space (not shown) and more effectively clusters the neurons. [sent-39, score-0.887]

30 1 Model Construction Dictionary learning We initially assume that spike detection has been performed on all channels. [sent-42, score-0.523]

31 , C} is a vector xn 2 RD , deﬁned by D time samples for each spike, centered at the peak of the detected signal; there are Nc spikes on channel c. [sent-49, score-0.604]

32 (c) Data from spike n on channel c, xn , is represented in terms of a dictionary D 2 RD⇥K , where K is an upper bound on the number of needed dictionary elements (columns of D), and the model 2 (c) infers the subset of dictionary elements needed to represent the data. [sent-50, score-1.467]

33 Each xn is represented as x(c) = D⇤(c) s(c) + ✏(c) n n n (c) (c) (1) (c) where ⇤(c) = diag( 1 b1 , 2 b2 , . [sent-51, score-0.039]

34 Deﬁning dk as the kth column of D, and letting ID represent the D ⇥ D identity matrix, the priors on the model parameters are dk ⇠ N (0, 1 ID ) , D (c) (c) k ⇠ T N + (0, c 1 ), ✏(c) ⇠ N (0, ⌃c 1 ) n (2) (c) where ⌃c = diag(⌘1 , . [sent-58, score-0.101]

35 Gamma priors (detailed when presenting results) are placed on c and on each of the ele(c) (c) ments of (⌘1 , . [sent-62, score-0.039]

36 For the binary vector b we impose the prior bk ⇠ Bernoulli(⇡k ), with ⇡k ⇠ Beta(a/K, b(K 1)/K), implying that the number of non-zero components of b is drawn Binomial(K, a/(a + b(K 1))); this corresponds to Poisson(a/b) in the limit K ! [sent-66, score-0.087]

37 (c) This model imposes that each xn is drawn from a linear subspace, deﬁned by the columns of D with corresponding non-zero components in b; the same linear subspace is shared across all channels (c) c 2 {1, . [sent-69, score-0.295]

38 However, the strength with which a column of D contributes toward xn depends on the channel c, as deﬁned by ⇤(c) . [sent-73, score-0.221]

39 Concerning ⇤(c) , rather than explicitly imposing a sparse (c) diagonal via b, we may also draw k ⇠ T N + (0, ck1 ), with shrinkage priors employed on the ck (i. [sent-74, score-0.147]

40 , with the ck drawn from a gamma prior that favors large ck ; which encourages many of the diagonal elements of ⇤(c) to be small, but typically not exactly zero). [sent-76, score-0.162]

41 In tests, the model performed similarly when shrinkage priors were used on ⇤(c) relative to explicit imposition of sparseness via b; all results below are based on the latter construction. [sent-77, score-0.118]

42 2 Multi-Channel Dynamic hierarchical Dirichlet process (c) We sort the spikes on the channels by clustering the {sn }, and in this sense feature design (learning {D⇤(c) }) and sorting are performed simultaneously. [sent-79, score-0.975]

43 We ﬁrst discuss how this may be performed via a hierarchical Dirichlet process (HDP) construction [20], and then extend this via a dynamic (c) HDP (dHDP) [17] considering multiple channels. [sent-80, score-0.216]

44 In an HDP construction, the {sn } are modeled as being drawn (c) s(c) ⇠ f (✓n ) , n (c) ✓n ⇠ G(c) , G(c) ⇠ DP(↵c G) , G ⇠ DP(↵0 G0 ) (3) P1 ⇤ where a Q from, for example, DP(↵0 G0 ) may be constructed [19] as G = i=1 ⇡i ✓i , where draw ⇤ ⇤ ⇤ ⇡i = Vi h <. [sent-81, score-0.028]

45 5ms) to an intracellular spike, we assume that the spike detected in the extracellular recording corresponds to the known neuron’s spikes. [sent-82, score-0.926]

46 This allows us to know partial ground truth, and allows us to test on methods compared to the known information. [sent-83, score-0.029]

47 For the accuracy analysis, we determine one cluster that corresponds to the known neuron. [sent-84, score-0.076]

48 Then we consider a spike to be correctly sorted if it is a known spike and is in the known cluster or if it is an unknown spike in the unknown cluster. [sent-85, score-1.492]

49 This data consists of a 4-channel extracellular recordings and 1-channel intracellular recording. [sent-87, score-0.283]

50 We used 2491 detected spikes and 786 of those spikes came from the known neuron. [sent-88, score-0.668]

51 The results show that learning the feature space instead of using the top 2 PCA components increases sorting accuracy. [sent-90, score-0.391]

52 This phenomenon can be seen in Figure 1, where it is impossible to accurately resolve the clusters in the space based on the 2 principal components, through either K-means or GMM. [sent-91, score-0.159]

53 Thus, by jointly learning the suitable feature space and clustering, we are able to separate the unknown and known neurons clusters more accurately. [sent-92, score-0.269]

54 In the HDP model the advantage is clear in the global accuracy as we achieve 89. [sent-93, score-0.042]

55 In addition to learning the appropriate feature space, HDP-DL and DP-DL can infer the appropriate number of clusters, allowing the data to deﬁne the number of neurons. [sent-96, score-0.049]

56 The posterior distribution on the number of global clusters and number of factors (dictionary elements) used is shown in Figure 3(a) and 3(b), along with the most used elements of the learned dictionary in Figure 3(c). [sent-97, score-0.414]

57 The dictionary elements show shapes similar to both neuron spikes in Figure 3(d) and wavelets. [sent-98, score-0.719]

58 Next we used the d561102 data from hc-1, which consists of 4 extracellular recording and 1 intracellular recording. [sent-100, score-0.356]

59 To do spike detection we high-pass ﬁltered the data from 300 Hz and detected spikes when the voltage level passed a positive or negative threshold, as in [2]. [sent-101, score-0.855]

60 We choose this data the known neuron displays dynamic properties by showing periods of activity and inactivity. [sent-102, score-0.295]

61 The intracellular recording in Figure 4(a) shows the known neuron is active for only a brief section of the recorded signal, and is then inactive for the rest of the signal. [sent-103, score-0.454]

62 The nonstationarity passes along to the extracellular spike train and the detect spikes. [sent-104, score-0.591]

63 We used the ﬁrst 930 detected spikes, which included 202 spikes from the known cluster. [sent-105, score-0.383]

64 In order to model the dynamic properties, we binned the data into 31 subgroups of 30 spikes to use with our multichannel dynamic HDP. [sent-106, score-0.559]

65 (a) approximate posterior probability on the number of global clusters (across all channels); (b) approximate posterior distribution on the number of dictionary elements; (c) six most used dictionary elements; (d) examples of typical spikes from the data. [sent-118, score-0.861]

66 The model adapts to the nonstationary spike dynamics by learning the parameters to model (c) dynamic properties at block 11 (w11 ⇡ 1, indicating that the dHDP has detected a change in the characteristics of the spikes), where the known neuron goes inactive. [sent-164, score-0.946]

67 Thus, the model is more likely to draw new local clusters at this point, reﬂecting the nonstationary data. [sent-165, score-0.182]

68 Additionally, in Figure 4(c) the global cluster usage shows a dramatic change at time block 11, where a cluster in the model goes inactive at the same time the known neuron is inactive. [sent-166, score-0.531]

69 Because the dynamic model can map these dynamic properties, the results improve while using this model. [sent-167, score-0.18]

70 Additionally, we obtain a global accuracy (across all channels) of 82. [sent-168, score-0.042]

71 66% using the HDP-DL and an global accuracy of 84. [sent-169, score-0.042]

72 1 The probability of introducing a new component for the 11th block 0. [sent-173, score-0.035]

73 2 0 10 20 Block Index 30 (d) Figure 4: Results of the multichannel dHDP on d561102. [sent-177, score-0.094]

74 (a) ﬁrst 40 seconds of the intracellular recording of d561102; (b) local cluster usage by each spike in the d561102 data in channel 4; (c) global cluster usage at (c) different time blocks for the data d561102; (d) sharing weight wj at each time blocks in the fourth channel. [sent-178, score-1.185]

75 The spike in 11 occurs when the known neuron goes inactive. [sent-179, score-0.681]

76 5 Conclusions We have presented a new method for performing multi-channel spike sorting, in which the underlying features (dictionary elements) and sorting are performed jointly, while also allowing timeevolving variation in the spike statistics. [sent-180, score-1.297]

77 The model adaptively learns dictionary elements of a wavelet-like nature (but not orthogonal), with characteristics like the shape of the spikes. [sent-181, score-0.289]

78 Kalman ﬁlter mixture model for spike sorting of non-stationary data. [sent-196, score-0.804]

79 Probabilistic inference of arm motion from neural activity in motor cortex. [sent-212, score-0.095]

80 Intracellular feautures predicted by extracellular recordings in the hippocampus in vivo. [sent-240, score-0.153]

81 A review of methods for spike sorting: the detection and classiﬁcation of neural action potentials. [sent-284, score-0.502]

82 Brain-machine interfaces to restore motor function and probe neural circuits. [sent-297, score-0.107]

83 Retrospective Markov Chain Monte Carlo methods for Dirichlet process hierarchiacal models. [sent-303, score-0.028]

84 Improved spike-sorting by modeling ﬁring statistics and burst-dependent spike amplitude attenuation: A Markov Chain Monte Carlo approach. [sent-310, score-0.472]

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The original SRM could account for subthreshold resonance, refractory effects and spike-frequency adaptation [11]. Mathematically similar models were developed independently in the study of the visual system [12] where spike-frequency adaptation has also been modeled [13]. Recently, this approach has retained increased attention since the probabilistic framework can be linked with the Bayesian theory of neural systems [14] and because Bayesian inference can be applied to the population of neurons [15]. In this paper, we investigate the similarity and differences between the state-of-the-art GLM and the stochastic AdEx. The motivation behind this work is to relate the traditional threshold neuron models to Bayesian theory. Our results extend the work of Plesser and Gerstner (2000) [16] since we include the non-linearity for spike initiation and spike-frequency adaptation. We also provide relationships between the parameters of the AdEx and the equivalent GLM. These precise relationships can be used to relate analog implementations of threshold models [17] to the probabilistic models used in the Bayesian approach. The paper is organized as follows: We ﬁrst describe the expressions relating the SRM state-variable to the parameters of the AdEx (Sect. 3.1) in the subthreshold regime. Then, we use numerical methods to ﬁnd the non-linear link-function that models the ﬁring probability (Sect. 3.2). We ﬁnd a functional form for the SRM link-function that best describes the ﬁring probability of a stochastic AdEx. We then compare the performance of this link-function with the often used exponential or linear-rectiﬁer link-functions (also called half-wave linear rectiﬁer) in terms of predicting the ﬁring probability of an AdEx under complex stimulus (Sect. 3.3). We ﬁnd that the exponential linkfunction yields almost perfect prediction. Finally, we explore the relations between the statistic of the noise and the sharpness of the non-linearity for spike initiation with the parameters of the SRM. 2 Presentation of the Models In this section we present the general formula for the stochastic AdEx model (Sect. 2.1) and the SRM (Sect 2.2). 2.1 The Stochastic Adaptive Exponential Integrate-and-Fire Model The voltage dynamics of the stochastic AdEx is given by: V −Θ ˙ τm V = El − V + ∆T exp − Rw + RI + R (1) ∆T τw w = a(V − El ) − w ˙ (2) where τm is the membrane time constant, El the reverse potential, R the membrane resistance, Θ is the threshold, ∆T is the shape factor and I(t) the input current which is chosen to be an Ornstein−Θ Uhlenbeck process with correlation time-constant of 5 ms. The exponential term ∆T exp( V∆T ) is a non-linear function responsible for the emission of spikes and is a diffusive white noise with standard deviation σ (i.e. ∼ N (0, σ)). Note that the diffusive white-noise does not imply white noise ﬂuctuations of the voltage V (t), the probability distribution of V (t) will depend on ∆T and Θ. The second variable, w, describes the subthreshold as well as the spike-triggered adaptation both ˆ parametrized by the coupling strength a and the time constant τw . Each time tj the voltage goes to inﬁnity, we assumed that a spike is emitted. Then the voltage is reset to a ﬁxed value Vr and w is increased by a constant value b. 2.2 The Generalized Linear Model In the SRM, The voltage V (t) is given by the convolution of the injected current I(t) with the membrane ﬁlter κ(t) plus the additional kernel η(t) that acts after each spikes (here we split the 2 spike-triggered kernel in two η(t) = ηv (t) + ηw (t) for reasons that will become clear later): V (t) = ˆ ˆ ηv (t − tj ) + ηw (t − tj ) El + [κ ∗ I](t) + (3) ˆ {tj } ˆ Then at each time tj a spike is emitted which results in a change of voltage described by η(t) = ηv (t) + ηw (t). Given the deterministic voltage, (Eq. 3) a spike is emitted according to the ﬁring intensity λ(V ): λ(t) = f (V (t)) (4) where f (·) is an arbitrary function called the link-function. Then the ﬁring behavior of the SRM depends on the choice of the link-function and its parameters. The most common link-function used to model single neuron activities are the linear-rectiﬁer and the exponential function. 3 Mapping In order to map the stochastic AdEx to the SRM we follow a two-step procedure. First we derive the ﬁlter κ(t) and the kernels ηv (t) and ηw (t) analytically as a function of AdEx parameters. Second, we derive the link-function of the SRM from the stochastic spike emission of the AdEx. Figure 1: Mapping of the subthreshold dynamics of an AdEx to an equivalent SRM. A. Membrane ﬁlter κ(t) for three different sets of parameters of the AdEx leading to over-damped, critically damped and under-damped cases (upper, middle and lower panel, respectively). B. Spike-Triggered η(t) (black), ηv (t) (light gray) and ηw (gray) for the three cases. C. Example of voltage trace produced when an AdEx is stimulated with a step of colored noise (black). The corresponding voltage from a SRM stimulated with the same current and where we forced the spikes to match those of the AdEx (red). D. Error in the subthreshold voltage (VAdEx − VGLM ) as a function of the mean voltage of the AdEx, for the three different cases: over-, critically and under-damped (light gray, gray and black, respectively) with ∆T = 1 mV. Red line represents the voltage threshold Θ. E. Root Mean Square Error (RMSE) ratio for the three cases with ∆T = 1 mV. The RMSE ratio is the RMSE between the deterministic VSRM and the stochastic VAdEx divided by the RMSE between repetitions of the stochastic AdEx voltage. The error bar shows a single standard deviation as the RMSE ratio is averaged accross multiple value of σ. 3.1 Subthreshold voltage dynamics We start by assuming that the non-linearity for spike initiation does not affect the mean subthreshold voltage of the stochastic AdEx (see Figure 1 D). This assumption is motivated by the small ∆T 3 observed in in-vitro recordings (from 0.5 to 2 mV [8, 9]) which suggest that the subthreshold dynamics are mainly linear except very close to Θ. Also, we expect that the non-linear link-function will capture some of the dynamics due to the non-linearity for spike initiation. Thus it is possible to rewrite the deterministic subthreshold part of the AdEx (Eq. 1-2 without and without ∆T exp((V − Θ)/∆T )) using matrices: ˙ x = Ax (5) with x = V w and A = − τ1 m a τw − gl1m τ − τ1 w (6) In this form, the dynamics of the deterministic AdEx voltage is a damped oscillator with a driving force. Depending on the eigenvalues of A the system could be over-damped, critically damped or under-damped. The ﬁlter κ(t) of the GLM is given by the impulse response of the system of coupled differential equations of the AdEx, described by Eq. 5 and 6. In other words, one has to derive the response of the system when stimulating with a Dirac-delta function. The type of damping gives three different qualitative shapes of the kernel κ(t), which are summarized in Table 3.1 and Figure 1 A. Since the three different ﬁlters also affect the nature of the stochastic voltage ﬂuctuations, we will keep the distinction between over-damped, critically damped and under-damped scenarios throughout the paper. This means that our approach is valid for at least 3 types of diffusive voltage-noise (i.e. the white noise in Eq. 1 ﬁltered by 3 different membrane ﬁlters κ(t)). To complete the description of the deterministic voltage, we need an expression for the spiketriggered kernels. The voltage reset at each spike brings a spike-triggered jump in voltage of magˆ nitude ∆ = Vr − V (t). This perturbation is superposed to the current ﬂuctuations due to I(t) and can be mediated by a Delta-diract pulse of current. Thus we can write the voltage reset kernel by: ηv (t) = ∆ ∆ [δ ∗ κ] (t) = κ(t) κ(0) κ(0) (7) where δ(t) is the Dirac-delta function. The shape of this kernel depends on κ(t) and can be computed from Table 3.1 (see Figure 1 B). Finally, the AdEx mediates spike-frequency adaptation by the jump of the second variables w. From Eq. 2 we can see that this produces a current wspike (t) = b exp (−t/τw ) that can cumulate over subsequent spikes. The effect of this current on voltage is then given by the convolution of wspike (t) with the membrane ﬁlter κ(t). Thus in the SRM framework the spike-frequency adaptation is taken into account by: ηw (t) = [wspike ∗ κ](t) (8) Again the precise form of ηw (t) depends on κ(t) and can be computed from Table 3.1 (see Figure 1 B). At this point, we would like to verify our assumption that the non-linearity for spike emission can be neglected. Fig. 1 C and D shows that the error between the voltage from Eq. 3 and the voltage from the stochastic AdEx is generally small. Moreover, we see that the main contribution to the voltage prediction error is due to the mismatch close to the spikes. However the non-linearity for spike initiation may change the probability distribution of the voltage ﬂuctuations, which in turn inﬂuences the probability of spiking. This will inﬂuence the choice of the link-function, as we will see in the next section. 3.2 Spike Generation Using κ(t), ηv (t) and ηw (t), we must relate the spiking probability of the stochastic AdEx as a function of its deterministic voltage. According to [2] the probability of spiking in time bin dt given the deterministic voltage V (t) is given by: p(V ) = prob{spike in [t, t + dt]} = 1 − exp (−f (V (t))dt) (9) where f (·) gives the ﬁring intensity as a function of the deterministic V (t) (Eq. 3). Thus to extract the link-function f we have to compute the probability of spiking given V (t) for our SRM. To do so we apply the method proposed by Jolivet et al. (2004) [18], where the probability of spiking is simply given by the distribution of the deterministic voltage estimated at the spike times divided by the distribution of the SRM voltage when there is no spike (see ﬁgure 2 A). One can numerically compute these two quantities for our models using N repetitions of the same stimulus. 4 Table 1: Analytical expressions for the membrane ﬁlter κ(t) in terms of the parameters of the AdEx for over-, critically-, and under-damped cases. Membrane Filter: κ(t) over-damped if: (τm + τw )2 > 4τm τw (gl +a) gl κ(t) = k1 eλ1 t + k2 eλ2 t λ1 = 1 2τm τw (−(τm + τw ) + critically-damped if: (τm + τw )2 = 4τm τw (gl +a) gl κ(t) = (αt + β)eλt λ= under-damped if: (τm + τw )2 < 4τm τw (gl +a) gl κ(t) = (k1 cos (ωt) + k2 sin (ωt)) eλt −(τm +τw ) 2τm τw λ= −(τm +τw ) 2τm τw (τm + τw )2 − 4 τm τw (gl + a) gl λ2 = 1 2τm τw (−(τm + τw ) − α= τm −τw 2Cτm τw ω= τw −τm 2τm τw 2 − a g l τm τw (τm + τw )2 − 4 τm τw (gl + a) gl k1 = −(1+(τm λ2 )) Cτm (λ1 −λ2 ) k2 = 1+(τm λ1 ) Cτm (λ1 −λ2 ) β= 1 C k1 = k2 = 1 C −(1+τm λ) Cωτm The standard deviation σ of the noise and the parameter ∆T of the AdEx non-linearity may affect the shape of the link-function. We thus extract p(V ) for different σ and ∆T (Fig. 2 B). Then using visual heuristics and previous knowledge about the potential analytical expression of the link-funtion, we try to ﬁnd a simple analytical function that captures p(V ) for a large range of combinations of σ and ∆T . We observed that the log(− log(p)) is close to linear in most studied conditions Fig. 2 B suggesting the following two distributions of p(V ): V − VT (10) p(V ) = 1 − exp − exp ∆V V − VT p(V ) = exp − exp − (11) ∆V Once we have p(V ), we can use Eq. 4 to obtain the equivalent SRM link-function, which leads to: −1 f (V ) = log (1 − p(V )) (12) dt Then the two potential link-functions of the SRM can be derived from Eq. 10 and Eq. 11 (respectively): V − VT f (V ) = λ0 exp (13) ∆V V − VT (14) f (V ) = −λ0 log 1 − exp − exp − ∆V 1 with λ0 = dt , VT the threshold of the SRM and ∆V the sharpness of the link-function (i.e. the parameters that governs the degree of the stochasticity). Note that the exact value of λ0 has no importance since it is redundant with VT . Eq. 13 is the standard exponential link-function, but we call Eq. 14 the log-exp-exp link-function. 3.3 Prediction The next point is to evaluate the ﬁt quality of each link-function. To do this, we ﬁrst estimate the parameters VT and ∆V of the GLM link-function that maximize the likelihood of observing a spike 5 Figure 2: SRM link-function. A. Histogram of the SRM voltage at the AdEx ﬁring times (red) and at non-ﬁring times (gray). The ratio of the two distributions gives p(V ) (Eq. 9, dashed lines). Inset, zoom to see the voltage histogram evaluated at the ﬁring time (red). B. log(− log(p)) as a function of the SRM voltage for three different noise levels σ = 0.07, 0.14, 0.18 nA (pale gray, gray, black dots, respectively) and ∆T = 1 mV. The line is a linear ﬁt corresponding to the log-exp-exp linkfunction and the dashed line corresponds to a ﬁt with the exponential link-function. C. Same data and labeling scheme as B, but plotting f (V ) according to Eq. 12. The lines are produced with Eq. 14 with parameters ﬁtted as described in B. and the dashed lines are produced with Eq. 13. Inset, same plot but on a semi-log(y) axis. train generated with an AdEx. Second we look at the predictive power of the resulting SRM in terms of Peri-Stimulus Time Histogram (PSTH). In other words we ask how close the spike trains generated with a GLM are from the spike train generated with a stochastic AdEx when both models are stimulated with the same input current. For any GLM with link-function f (V ) ≡ f (t|I, θ) and parameters θ regulating the shape of κ(t), ˆ ηv (t) and ηw (t), the Negative Log-Likelihood (NLL) of observing a spike-train {t} is given by:   NLL = − log(f (t|I, θ)) − f (t|I, θ) (15) t ˆ t It has been shown that the negative log-likelihood is convex in the parameters if f is convex and logconcave [19]. It is easy to show that a linear-rectiﬁer link-function, the exponential link-function and the log-exp-exp link-function all satisfy these conditions. This allows efﬁcient estimation of ˆ ˆ the optimal parameters VT and ∆V using a simple gradient descent. One can thus estimate from a single AdEx spike train the optimal parameters of a given link-function, which is more efﬁcient than the method used in Sect. 3.2. The minimal NLL resulting from the gradient descent gives an estimation of the ﬁt quality. A better estimate of the ﬁt quality is given by the distance between the PSTHs in response to stimuli not used for parameter ﬁtting . Let ν1 (t) be the PSTH of the AdEx, and ν2 (t) be the PSTH of the ﬁtted SRM, 6 Figure 3: PSTH prediction. A. Injected current. B. Voltage traces produced by an AdEx (black) and the equivalent SRM (red), when stimulated with the current in A. C. Raster plot for 20 realizations of AdEx (black tick marks) and equivalent SRM (red tick marks). D. PSTH of the AdEx (black) and the SRM (red) obtained by averaging 10,000 repetitions. E. Optimal log-likelihood for the three cases of the AdEx, using three different link-functions, a linear-rectiﬁer (light gray), an exponential link-function (gray) and the link-function deﬁned by Eq. 14 (dark gray), these values are obtained by averaging over 40 different combinations σ and ∆T (see Fig. 4). Error bars are one standard deviation, the stars denote a signiﬁcant difference, two-sample t-test with α = 0.01. F. same as E. but for Md (Eq. 16). then we use Md ∈ [0, 1] as a measure of match: Md = 2 2 (ν1 (t) − ν2 (t)) dt ν1 (t)2 dt + ν2 (t)2 dt (16) Md = 1 means that it is impossible to differentiate the SRM from the AdEx in terms of their PSTHs, whereas a Md of 0 means that the two PSTHs are completely different. Thus Md is a normalized similarity measure between two PSTHs. In practice, Md is estimated from the smoothed (boxcar average of 1 ms half-width) averaged spike train of 1 000 repetitions for each models. We use both the NLL and Md to quantify the ﬁt quality for each of the three damping cases and each of the three link-functions. Figure 3 shows the match between the stochastic AdEx used as a reference and the derived GLM when both are stimulated with the same input current (Fig. 3 A). The resulting voltage traces are almost identical (Fig. 3 B) and both models predict almost the same spike trains and so the same PSTHs (Fig. 3 C and D). More quantitalively, we see on Fig. 3 E and F, that the linear-rectiﬁer ﬁts signiﬁcantly worse than both the exponential and log-exp-exp link-functions, both in terms of NLL and of Md . The exponential link-function performs as well as the log-exp-exp link-function, with a spike train similarity measure Md being almost 1 for both. Finally the likelihood-based method described above gives us the opportunity to look at the relationship between the AdEx parameters σ and ∆T that governs its spike emission and the parameters VT and ∆V of the link-function (Fig. 4). We observe that an increase of the noise level produces a ﬂatter link-function (greater ∆V ) while an increase in ∆T also produces an increase in ∆V and VT (note that Fig. 4 shows ∆V and VT for the exponential link-function only, but equivalent results are obtained with the log-exp-exp link-function). 4 Discussion In Sect. 3.3 we have shown that it is possible to predict with almost perfect accuracy the PSTH of a stochastic AdEx model using an appropriate set of parameters in the SRM. Moreover, since 7 Figure 4: Inﬂuence of the AdEx parameters on the parameters of the exponential link-function. A. VT as a function of ∆T and σ. B. ∆V as a function of ∆T and σ. the subthreshold voltage of the AdEx also gives a good match with the deterministic voltage of the SRM, we expect that the AdEx and the SRM will not differ in higher moments of the spike train probability distributions beyond the PSTH. We therefore conclude that diffusive noise models of the type of Eq. 1-2 are equivalent to GLM of the type of Eq. 3-4. Once combined with similar results on other types of stochastic LIF (e.g. correlated noise), we could bridge the gap between the literature on GLM and the literature on diffusive noise models. Another noteworthy observation pertains to the nature of the link-function. The link-function has been hypothesized to be a linear-rectiﬁer, an exponential, a sigmoidal or a Gaussian [16]. We have observed that for the AdEx the link-function follows Eq. 14 that we called the log-exp-exp linkfunction. Although the link-function is log-exp-exp for most of the AdEx parameters, the exponential link-function gives an equivalently good prediction of the PSTH. This can be explained by the fact that the difference between log-exp-exp and exponential link-functions happens mainly at low voltage (i.e. far from the threshold), where the probability of emitting a spike is so low (Figure 2 C, until -50 mv). Therefore, even if the exponential link-function overestimates the ﬁring probability at these low voltages it rarely produces extra spikes. At voltages closer to the threshold, where most of the spikes are emitted, the two link-functions behave almost identically and hence produce the same PSTH. The Gaussian link-function can be seen as lying in-between the exponential link-function and the log-exp-exp link-function in Fig. 2. This means that the work of Plesser and Gerstner (2000) [16] is in agreement with the results presented here. The importance of the time-derivative of the ˙ voltage stressed by Plesser and Gerstner (leading to a two-dimensional link-function f (V, V )) was not studied here to remain consistent with the typical usage of GLM in neural systems [14]. Finally we restricted our study to exponential non-linearity for spike initiation and do not consider other cases such as the Quadratic Integrate-and-ﬁre (QIF, [5]) or other polynomial functional shapes. We overlooked these cases for two reasons. First, there are many evidences that the non-linearity in neurons (estimated from in-vitro recordings of Pyramidal neurons) is well approximated by a single exponential [9]. Second, the exponential non-linearity of the AdEx only affects the subthreshold voltage at high voltage (close to threshold) and thus can be neglected to derive the ﬁlters κ(t) and η(t). Polynomial non-linearities on the other hand affect a larger range of the subthreshold voltage so that it would be difﬁcult to justify the linearization of subthreshold dynamics essential to the method presented here. References [1] R. B. Stein, “Some models of neuronal variability,” Biophys J, vol. 7, no. 1, pp. 37–68, 1967. [2] W. Gerstner and W. Kistler, Spiking neuron models. Cambridge University Press New York, 2002. [3] E. Izhikevich, “Resonate-and-ﬁre neurons,” Neural Networks, vol. 14, no. 883-894, 2001. [4] M. J. E. Richardson, N. Brunel, and V. Hakim, “From subthreshold to ﬁring-rate resonance,” Journal of Neurophysiology, vol. 89, pp. 2538–2554, 2003. 8 [5] E. Izhikevich, “Simple model of spiking neurons,” IEEE Transactions on Neural Networks, vol. 14, pp. 1569–1572, 2003. [6] S. Mensi, R. Naud, M. Avermann, C. C. H. Petersen, and W. Gerstner, “Parameter extraction and classiﬁcation of three neuron types reveals two different adaptation mechanisms,” Under review. [7] N. Fourcaud-Trocme, D. Hansel, C. V. Vreeswijk, and N. Brunel, “How spike generation mechanisms determine the neuronal response to ﬂuctuating inputs,” Journal of Neuroscience, vol. 23, no. 37, pp. 11 628–11 640, 2003. [8] R. Brette and W. Gerstner, “Adaptive exponential integrate-and-ﬁre model as an effective description of neuronal activity,” Journal of Neurophysiology, vol. 94, pp. 3637–3642, 2005. [9] L. Badel, W. Gerstner, and M. Richardson, “Dependence of the spike-triggered average voltage on membrane response properties,” Neurocomputing, vol. 69, pp. 1062–1065, 2007. [10] P. McCullagh and J. A. Nelder, Generalized linear models, 2nd ed. Chapman & Hall/CRC, 1998, vol. 37. [11] W. Gerstner, J. van Hemmen, and J. Cowan, “What matters in neuronal locking?” Neural computation, vol. 8, pp. 1653–1676, 1996. [12] D. Hubel and T. Wiesel, “Receptive ﬁelds and functional architecture of monkey striate cortex,” Journal of Physiology, vol. 195, pp. 215–243, 1968. [13] J. Pillow, L. Paninski, V. Uzzell, E. Simoncelli, and E. Chichilnisky, “Prediction and decoding of retinal ganglion cell responses with a probabilistic spiking model,” Journal of Neuroscience, vol. 25, no. 47, pp. 11 003–11 013, 2005. [14] K. Doya, S. Ishii, A. Pouget, and R. P. N. Rao, Bayesian brain: Probabilistic approaches to neural coding. The MIT Press, 2007. [15] S. Gerwinn, J. H. Macke, M. Seeger, and M. Bethge, “Bayesian inference for spiking neuron models with a sparsity prior,” in Advances in Neural Information Processing Systems, 2007. [16] H. Plesser and W. Gerstner, “Noise in integrate-and-ﬁre neurons: From stochastic input to escape rates,” Neural Computation, vol. 12, pp. 367–384, 2000. [17] J. Schemmel, J. Fieres, and K. Meier, “Wafer-scale integration of analog neural networks,” in Neural Networks, 2008. IJCNN 2008. (IEEE World Congress on Computational Intelligence). IEEE International Joint Conference on, june 2008, pp. 431 –438. [18] R. Jolivet, T. Lewis, and W. Gerstner, “Generalized integrate-and-ﬁre models of neuronal activity approximate spike trains of a detailed model to a high degree of accuracy,” Journal of Neurophysiology, vol. 92, pp. 959–976, 2004. [19] L. Paninski, “Maximum likelihood estimation of cascade point-process neural encoding models,” Network: Computation in Neural Systems, vol. 15, pp. 243–262, 2004. 9

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