Neural spike statistics modify the impact of background noise

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1 Neurocomputing 38}40 (2001) 445}450 Neural spike statistics modify the impact of background noise Stefan D. Wilke*, Christian W. Eurich Institut fu( r Theoretische Physik, Universita( t Bremen, Postfach , D Bremen, Germany Abstract Neural populations in the neocortex typically encode multiple stimulus features, e.g., position, brightness, contrast, and orientation of a visual stimulus in the case of cells in area 17. Here, we perform a Fisher information analysis of the encoding accuracy of a neural population which is sensitive to D stimulus features. The neurons are assumed to exhibit a non-vanishing level of baseline activity. It is shown that the encoding accuracy decreases drastically with D if the spike count variance depends on the mean spike count, as is the case for Poissonian spike statistics. The need to reduce the susceptibility to background noise thus poses severe restrictions on the neural "ring statistics or the number of encoded stimulus features. The results hold for uncorrelated as well as for correlated activity in the neural population Elsevier Science B.V. All rights reserved. PACS: La, !r Keywords: Population coding; Response variability; Baseline activity; Fisher information 1. Introduction The neocortex consists of dozens of areas whose neurons process di!erent*although sometimes overlapping*features of sensory stimuli. This diversi"cation of signal processing poses the problem of binding the di!erent features belonging to a single object in order to obtain an appropriate cortical object representation [8]. This problem could be avoided by neurons that are equally sensitive to all, or at least to many, stimulus features. The observation that this is not the case leads to the * Corresponding author. addresses: swilke@physik.uni-bremen.de (S.D. Wilke), eurich@physik.uni-bremen.de (C.W. Eurich) /01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S ( 0 1 )

2 446 S.D. Wilke, C.W. Eurich / Neurocomputing 38}40 (2001) 445}450 question why neurons in many areas are specialized in a restricted number of stimulus features. Part of the answer lies in the fact that the localization of a stimulus de"ned by a large number of features requires the neural population to cover a stimulus space of correspondingly high dimension. The number of neurons necessary for the representation of D features scales as a, where a is a measure for the acuity to be obtained in each dimension. Thus, there is a combinatorial explosion in the population size. However, the fact that large receptive "elds yield a high representational accuracy [10,4,13] may restrict the number of required neurons. Here, we relate the problem of the number of encoded dimensions to the spontaneous activity present in most cortical neurons. A Fisher information analysis [9,13,11,5,6,12] suggests that the neural population meets with a drastic loss of information content as D is increased. Satisfactory representational accuracy can be achieved through a restriction in D or through the presence of additive spike count noise. 2. Theoretical framework Suppose that a neural population of N neurons encodes a stimulus x"(x,2, x ) consisting of D features. Since the neurons "re in a stochastic manner, the response to a given stimulus is characterized by the joint probability distribution P(nx), where n"(n,2, n ) is the vector of spike counts measured within an observation time τ.in order to assess the coding accuracy achieved by the neural population, we employ the total Fisher information J(x), J(x):" 1 J (x), J (x):"e n x ln P(n x), (1) where E n [2] denotes the expectation value over the distribution P(n x), and J (x)is the Fisher information for the individual stimulus feature x. In the present context, the total Fisher information is especially suitable since its inverse is a lower bound for the squared error of an unbiased stimulus reconstruction (CrameH r}rao inequality, see e.g. [3]), E n [(x( (n)!x)]*1/j(x). This inequality holds for any unbiased spike-count based estimate x( (n) of the stimulus x. Correlations between the neural spike counts are described by assuming that P(nx) has the form of a Gaussian probability distribution with covariance matrix Q(x) [1]. A general factorization ansatz for this matrix was made by Zhang and Sejnowski [13], Q (x)"[δ #(1!δ )q]ψ[ f (x)]ψ[ f (x)] (2) with a parameter 1'q*0 measuring correlation strength, and an arbitrary function ψ. The vector f(x)"( f (x),2, f (x)) combines the mean "ring rates, i.e., f(x)"n/τ. As special cases, Eq. (2) covers additive noise (ψ(z),s with an arbitrary constant so0) [7], multiplicative noise (ψ(z)"sz), and independent Poissonian spiking in the

3 S.D. Wilke, C.W. Eurich / Neurocomputing 38}40 (2001) 445} limit Fτ<1 (ψ(z)"zτ and q"0). Note that, even in the absence of correlations (q"0), these three noise models are substantially di!erent in terms of their variance Q (x): The additive noise has constant variance Q (x)"s, while multiplicative and Poissonian noise have variances that increase with the mean, Q (x)"sf (x) and Q (x)"f (x)τ, respectively. The tuning functions f (x), k"1,2, N, are assumed to be of the form f (x)"f(ξ), with ξ :" ξ and ξ :"(x!c)/σ, where σ denotes the neuronal tuning width for feature i. This form implies that the mean "ring rate only depends on the sum of the squares of rescaled distances ξ between the stimulus x and the tuning curve center c. The tuning curve shape function is normalized to unity, so that F denotes the maximal "ring rate. Note that the tuning widths σ need not be equal, thus allowing broad tuning for some and narrow tuning for other stimulus features. If there is su$cient tuning curve overlap and N is large, the Fisher information for the model population is given by [11,13] with J"η σ (1/D) σ K (F, τ, D, q) (3 ) ( K (F, τ, D, q):" 1 ( 1!q A (F, τ, D)#2!q ( 1!q B (F, τ, D), (4) ( where η is the ("xed) number of tuning curves per unit volume in the stimulus space, and the functions A (F, τ, D) and B (F, τ, D) are factors that depend on the shape of ( ( the tuning curve and on the correlation model function ψ, but not on the tuning widths σ. They read as A ( (F, τ, D):" 4F D B ( (F, τ, D):" 4F D π Γ(1#(D/2)) π Γ(1#(D/2)) dξ (ξ)ξ ψ[f(ξ)], (5) dξ ψ[f(ξ)](ξ)ξ. (6) ψ[f(ξ)] As noted in [1,13], the e!ect of uniform correlations of the form of Eq. (2) on the population's Fisher information is a factor 1/(1!q) that increases the accuracy in the presence of positive correlations (q'0). 3. Results and discussion In order to incorporate background activity in the present model, suppose that the neurons' tuning functions f (x) consist of a stimulus response proportional to FI (where I is normalized to unity) and a constant baseline "ring rate level Fζ (0)ζ)1), F(z)"Fζ#F(1!ζ)I(z). (7)

4 448 S.D. Wilke, C.W. Eurich / Neurocomputing 38}40 (2001) 445}450 Fig. 1. Total Fisher information J as a function of baseline activity level ζ for di!erent numbers D of encoded features. The curves have been normalized to unity at ζ"0 in order to make them comparable. Curves for D"1, 2, 3, 10 were obtained assuming independent Poisson neurons, Eq. (8), the dot}dashed curve results from the additive noise model ψ"const., Eq. (9). For the population described in the previous section, the background activity implies that the terms A ( (F, τ, D) and B ( (F, τ, D) become functions of the baseline "ring level ζ. Assuming independent Poissonian "ring at high spike counts, Fτ<1, and Gaussian tuning curves, I(z)"exp(!z/2), one "nds A ( <B ( and K ( (F, τ, D, q"0)" Fτ D (2π) 1!ζ#ζ Li 1!1 ζ, (8) where Li (z):" z/k is the polylogarithm function. The plot of this function in Fig. 1 demonstrates that the baseline activity (as measured by ζ) deteriorates the encoding accuracy as expected. An interesting "nding is that the degree to which this is the case strongly depends on the number of stimulus features D. For large D, background activity appears to be disastrous for the encoding: At D"10, for example, a noise level of ζ"0.1 already leads to a 92% decrease of the Fisher information, which is equivalent to a more than threefold increase of the minimal reconstruction error. The underlying reason for this behavior is that, for high dimensionality, most of the volume of the receptive "eld falls into the `border regiona of the tuning curve (I;1), where the majority of spikes belong to the background activity. A qualitatively similar dependence on ζ and D results for other tuning curve shapes, e.g. cos-tuning.

5 S.D. Wilke, C.W. Eurich / Neurocomputing 38}40 (2001) 445} Eqs. (5)}(7) imply that the impact of background activity on Fisher information depends on the correlation model ψ that is adapted. Assuming additive noise (ψ(z),s), for example, one "nds K (F, τ, D, q)" 1 4F π(1!ζ) ( 1!q sdγ(1#(d/2)) dξi(ξ)ξ. (9) The above equation shows that the ζ-dependence for this kind of "ring rate variance is always proportional to (1!ζ), regardless of the dimension D. The corresponding normalized Fisher information is included in Fig. 1 (dot}dashed line). Apparently, it does not show the increase in susceptibility to background activity at large D, asis obvious from Eq. (9). In addition, its ζ-robustness is always superior to the Poisson model, even at D"1. Thus, a di!erent "ring rate variance can remedy the drastic decrease of Fisher information with the level of background activity for large D. 4. Conclusion Our Fisher information calculation of the encoding accuracy of a spontaneously active neural population suggests that system performance rapidly decreases with the number D of encoded stimulus features. This holds for usual types of noisy responses, where the variance of the spike count increases with the mean spike count. The problem can be avoided through a restriction in D, and does not occur for special types of noise, e.g. with constant variance. The analysis of empirical data from cortical neurons will shed further light on the connection between encoding accuracy, stimulus-uncorrelated background activity, and the number of relevant stimulus features. For example, the measurement of local "eld potentials in cat area 17 suggests the existence of background activity related to the dynamics of the underlying cortical network [2]. The information}theoretic consequences of this behavior remain to be elucidated. References [1] L.F. Abbott, P. Dayan, The e!ect of correlated variability on the accuracy of a population code, Neural Comput. 11 (1999) 91}101. [2] A. Arieli, A. Sterkin, A. Grinvald, A. Aertsen, Dynamics of ongoing activity: Explanation of the Large Variability in Evoked Cortical Responses, Science 273(1996) 1868}1871. [3] T.M. Cover, J.A. Thomas, Elements of Information Theory, Wiley, New York, [4] C.W. Eurich, H. Schwegler, Coarse coding: calculation of the resolution achieved by a population of large receptive "eld neurons, Biol. Cybernet. 76 (1997) 357}363. [5] C.W. Eurich, S.D. Wilke, Multi-dimensional encoding strategy of spiking neurons, Neural Comput. 12 (2000) 1519}1529. [6] C.W. Eurich, S.D. Wilke, H. Schwegler, Neural representation of multi-dimensional stimuli, in: S.A. Solla, T.K. Leen, K.-R. MuK ller (Eds.), Advances in Neural Information Processing Systems, Vol. 12, MIT Press, Cambridge, MA, 2000, pp. 115}121. [7] K.O. Johnson, Sensory discrimination: neural processes preceding discrimination decision, J. Neurophysiol. 43(1980) 1793}1815.

6 450 S.D. Wilke, C.W. Eurich / Neurocomputing 38}40 (2001) 445}450 [8] A.L. Roskies, The binding problem, Neuron 24 (1999) 7}9. [9] H.S. Seung, H. Sompolinski, Simple models for reading neuronal population codes, Proc. Natl. Acad. Sci. USA 90 (1993) 10749} [10] H.P. Snippe, J.J. Koenderink, Discrimination thresholds for channel-coded systems, Biol. Cybernet. 66 (1992) 543}551. [11] S.D. Wilke, C.W. Eurich, What does a neuron talk about?, in: M. Verleysen (Ed.), Proc. ESANN '99, D-Facto, Bruxelles, 1999, pp. 435}440. [12] S.D. Wilke, C.W. Eurich, Representational accuracy of stochastic neural populations, Neural Comput. (2000), submitted. [13] K. Zhang, T.J. Sejnowski, Neuronal tuning: to sharpen or broaden?, Neural Comput. 11 (1999) 75}84. Stefan D. Wilke studied physics at the University of Konstanz (Germany) and at the Freie UniversitaK t Berlin (Germany). In 1998, he "nished his Diploma thesis on mode-coupling theory of the glass transition in ionic #uids. Since 1998, he has been a Ph.D. student at Department of Neurophysics at the University of Bremen. His research interests are theoretical aspects of neural coding and early visual processing. Christian W. Eurich got his Ph.D. in Theoretical Physics in 1995 from the University of Bremen (Germany). As a postdoc, he worked with John Milton and Jack Cowan at the University of Chicago, and he was guest researcher at the Max-Planck Institut fuk r StroK mungsforschung in GoK ttingen (Germany) and at the RIKEN Brain Institute in Tokyo. Since 1997, he has been working at the Department of Theoretical Neurophysics at the University of Bremen. His research interests include neural networks with time delays, neural information processing and neural coding, visuomotor behavior, and psychophysical modelling.