nips nips2010 nips2010-96 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Jaldert Rombouts, Sander M. Bohte
Abstract: Recent experimental work has suggested that the neural firing rate can be interpreted as a fractional derivative, at least when signal variation induces neural adaptation. Here, we show that the actual neural spike-train itself can be considered as the fractional derivative, provided that the neural signal is approximated by a sum of power-law kernels. A simple standard thresholding spiking neuron suffices to carry out such an approximation, given a suitable refractory response. Empirically, we find that the online approximation of signals with a sum of powerlaw kernels is beneficial for encoding signals with slowly varying components, like long-memory self-similar signals. For such signals, the online power-law kernel approximation typically required less than half the number of spikes for similar SNR as compared to sums of similar but exponentially decaying kernels. As power-law kernels can be accurately approximated using sums or cascades of weighted exponentials, we demonstrate that the corresponding decoding of spiketrains by a receiving neuron allows for natural and transparent temporal signal filtering by tuning the weights of the decoding kernel. 1
Reference: text
sentIndex sentText sentNum sentScore
1 nl Abstract Recent experimental work has suggested that the neural firing rate can be interpreted as a fractional derivative, at least when signal variation induces neural adaptation. [sent-9, score-0.843]
2 Here, we show that the actual neural spike-train itself can be considered as the fractional derivative, provided that the neural signal is approximated by a sum of power-law kernels. [sent-10, score-1.063]
3 A simple standard thresholding spiking neuron suffices to carry out such an approximation, given a suitable refractory response. [sent-11, score-0.451]
4 Empirically, we find that the online approximation of signals with a sum of powerlaw kernels is beneficial for encoding signals with slowly varying components, like long-memory self-similar signals. [sent-12, score-0.942]
5 For such signals, the online power-law kernel approximation typically required less than half the number of spikes for similar SNR as compared to sums of similar but exponentially decaying kernels. [sent-13, score-0.595]
6 As power-law kernels can be accurately approximated using sums or cascades of weighted exponentials, we demonstrate that the corresponding decoding of spiketrains by a receiving neuron allows for natural and transparent temporal signal filtering by tuning the weights of the decoding kernel. [sent-14, score-1.234]
7 1 Introduction A key issue in computational neuroscience is the interpretation of neural signaling, as expressed by a neuron’s sequence of action potentials. [sent-15, score-0.026]
8 An emerging notion is that neurons may in fact encode information at multiple timescales simultaneously [1, 2, 3, 4]: the precise timing of spikes may be conveying high-frequency information, and slower measures, like the rate of spiking, may be relating low-frequency information. [sent-16, score-0.699]
9 Such multi-timescale encoding comes naturally, at least for sensory neurons, as the statistics of the outside world often exhibit self-similar multi-timescale features [5] and the magnitude of natural signals can extend over several orders. [sent-17, score-0.465]
10 Since neurons are limited in the rate and resolution with which they can emit spikes, the mapping of large dynamic-range signals into spike-trains is an integral part of attempts at understanding neural coding. [sent-18, score-0.461]
11 Experiments have extensively demonstrated that neurons adapt their response when facing persistent changes in signal magnitude. [sent-19, score-0.575]
12 Typically, adaptation changes the relation between the magnitude of the signal and the neuron’s discharge rate. [sent-20, score-0.433]
13 Since adaptation thus naturally relates to neural coding, it has been extensively scrutinized [6, 7, 8]. [sent-21, score-0.169]
14 Tying the notions of self-similar multi-scale natural signals and adaptive neural coding together, it has recently been suggested that neuronal adaptation allows neuronal spiking to communicate a fractional derivative of the actual computed signal [10, 4]. [sent-25, score-1.844]
15 Fractional derivatives are a generalization of standard ‘integer’ derivatives (‘first order’, ‘second order’), to real valued derivatives (e. [sent-26, score-0.261]
16 A key feature of such derivatives is that they are non-local, and rather convey information over essentially a large part of the signal spectrum [10]. [sent-30, score-0.402]
17 1 Here, we show how neural spikes can encode temporal signals when the spike-train itself is taken as the fractional derivative of the signal. [sent-31, score-1.4]
18 We show that this is the case for a signal approximated by a sum of shifted power-law kernels starting at respective times ti and decaying proportional to 1/(t − ti )β . [sent-32, score-0.853]
19 Then, the fractional derivative of this approximated signal corresponds to a sum of spikes at times ti , provided that the order of fractional differentiation α is equal to 1 − β: a spiketrain is the α = 0. [sent-33, score-2.098]
20 2 fractional derivative of a signal approximated by a sum of power-law kernels with exponent β = 0. [sent-34, score-1.347]
21 Such signal encoding with power-law kernels can be carried out for example with simple standard thresholding spiking neurons with a refractory reset following a power-law. [sent-36, score-0.997]
22 As fractional derivatives contain information over many time-ranges, they are naturally suited for predicting signals. [sent-37, score-0.61]
23 This links to notions of predictive coding, where neurons communicate deviations from expected signals rather than the signal itself. [sent-38, score-1.036]
24 Predictive coding has been suggested as a key feature of neuronal processing in e. [sent-39, score-0.182]
25 For self-similar scale-free signals, future signals may be influenced by past signals over very extended time-ranges: so-called longmemory. [sent-42, score-0.568]
26 For example, fractional Brownian motion (fBm) can exhibit long-memory, depending on their Hurst-parameter H. [sent-43, score-0.555]
27 5 fBM models which exhibit long-range dependence (longmemory) where the autocorrelation-function follows a power-law decay [12]. [sent-45, score-0.12]
28 The long-memory nature of signals approximated with sums of power-law kernels naturally extends this signal approximation into the future along the autocorrelation of the signal, at least for self-similar 1/f γ like signals. [sent-46, score-1.038]
29 The key “predictive” assumption we make is that a neuron’s spike-train up to time t contains all the information that the past signal contributes to the future signal t > t. [sent-47, score-0.614]
30 The correspondence between a spike-train as a fractional derivative and a signal approximated as a sum of power-law kernels is only exact when spike-trains are taken as a sum of Dirac-δ functions and the power-law kernels as 1/tβ . [sent-48, score-1.538]
31 As both responses are singular, neurons would only be able to approximate this. [sent-49, score-0.188]
32 We show empirically how sums of (approximated) 1/tβ power-law kernels can accurately approximate long-memory fBm signals via simple difference thresholding, in an online greedy fashion. [sent-50, score-0.5]
33 Thus encodings signals, we show that the power-law kernels approximate synthesized signals with about half the number of spikes to obtain the same Signal-to-Noise-Ratio, when compared to the same encoding method using similar but exponentially decaying kernels. [sent-51, score-1.026]
34 We further demonstrate the approximation of sine wave modulated white-noise signals with sums of power-law kernels. [sent-52, score-0.525]
35 We find the effect is stronger when encoding the actual sine wave envelope, mimicking the difference between thalamic and cortical neurons reported in [4]. [sent-54, score-0.636]
36 This may suggest that these cortical neurons are more concerned with encoding the sine wave envelope. [sent-55, score-0.507]
37 The power-law approximation also allows for the transparent and straightforward implementation of temporal signal filtering by a post-synaptic, receiving neuron. [sent-56, score-0.514]
38 Since neural decoding by a receiving neuron corresponds to adding a power-law kernel for each received spike, modifying this receiving power-law kernel then corresponds to a temporal filtering operation, effectively exploiting the wide-spectrum nature of power-law kernels. [sent-57, score-0.619]
39 This is particularly relevant, since, as has been amply noted [9, 14], power-law dynamics can be closely approximated by a weighted sum or cascade of exponential kernels. [sent-58, score-0.238]
40 Temporal filtering would then correspond to simply tuning the weights for this sum or cascade. [sent-59, score-0.06]
41 We illustrate this notion with an encoding/decoding example for both a high-pass and low-pass filter. [sent-60, score-0.03]
42 2 Power-law Signal Encoding Neural processing can often be reduced to a Linear-Non-Linear (LNL) filtering operation on incoming signals [15] (figure 1), where inputs are linearly weighted and then passed through a non-linearity to yield the neural activation. [sent-61, score-0.368]
43 As this computation yields analog activations, and neurons communicate through spikes, the additional problem faced by spiking neurons is to decode the incoming signal and then encode the computed LNL filter again into a spike-train. [sent-62, score-0.985]
44 The standard spiking neuron model is that of Linear-Nonlinear-Poisson spiking, where spikes have a stochastic relationship to the computed activation [16]. [sent-63, score-0.695]
45 Here, we interpret the spike encoding and decoding in the light of processing and communicating signals with fractional derivatives [10]. [sent-64, score-1.1]
46 and summing these kernels [17]; keeping track of doublets and triplet spikes allows for even greater fidelity. [sent-66, score-0.567]
47 This approach however only worked for signals with a frequency response lacking low frequencies [17]. [sent-67, score-0.347]
48 Low-frequency changes lead to “adaptation”, where the kernel is adapted to fit the signal again [18]. [sent-68, score-0.38]
49 For long-range predictive coding, the absence of low frequencies leaves little to predict, as the effective correlation time of the signals is then typically very short as well [17]. [sent-69, score-0.448]
50 Using the notion of predictive coding in the context of (possible) long-range dependencies, we define the goal of signal encoding as follows: let a signal xj (t) be the result of the continuous-time computation in neuron j up to time t, and let neuron j have emitted spikes tj up to time t. [sent-70, score-1.719]
51 These spikes should be emitted such that the signal xj (t ) for t < t is decoded up to some signal-to-noise ratio, and these spikes should be predictive for xj (t ) for t > t in the sense that no additional spikes are needed at times t > t to convey the predictive information up to time t. [sent-71, score-1.86]
52 Taking kernels as a signal filter of fixed width, as in the general approach in [17] has the important drawback that the signal reconstruction incurs a delay for the duration of the filter: its detection cannot be communicated until the filter is actually matched to the signal. [sent-72, score-0.784]
53 Alternatively, a predictive coding approach could rely on only on a very short backward looking filter, minimizing the delay in the system, and continuously computing a forward predictive signal. [sent-74, score-0.485]
54 At any time in the future then, only deviations of the actual signal from this expectation are communicated. [sent-75, score-0.372]
55 1 Spike-trains as fractional derivative As recent work has highlighted the possibility that neurons encode fractional derivatives, it is noteworthy that the non-local nature of fractional calculus offers a natural framework for predictive coding. [sent-77, score-2.102]
56 The fractional derivative r(t) of a signal x(t) is denoted as Dα x(t), and intuitively expresses: r(t) = dα x(t), dtα where α is the fractional order, e. [sent-79, score-1.416]
57 We assume that neurons carry out predictive coding by emitting spikes such that all predictive information is contained in the current spikes, and no more spikes will be fired if the signal follows this prediction. [sent-84, score-1.669]
58 Approximating spikes by Dirac-δ functions, we take the spike-train up to some time t0 to be the fractional derivative of the past signal and be fully predictive for the expected influence the 3 Fractionally Predicting Spikes a) x(t) r(t) 0 c) 0. [sent-85, score-1.507]
59 3 time (s) Non−singular kernels b) x(t) r(t) α-exp(τ=10ms) 0 0. [sent-88, score-0.172]
60 c) Approximated 1/tβ power-law kernel for different values of k from eq. [sent-95, score-0.06]
61 d) The approximated 1/tβ power-law kernel (blue line) can be decomposed as a weighted sum of α-functions with various decay time-constants (dashed lines). [sent-97, score-0.353]
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