THEORETICAL NEUROSCIENCE - PART 3 docx

2.6 Receptive Fields in the Retina and LGN 35 03 0 200 0 0 3 3 -2 -1 0 1 2 0 100 200 0 D xt -2 -1 0 1 2 -2 0 2 0 D s B D A C Figure 2.25: Receptive fields of LGN neurons. A) The center-surround spatial structure of the receptive field of a cat LGN X cell. This has a central ON region (solid contours) and a surrounding OFF region (dashed contours). B) A fitofthe receptive field shown in A using a difference of Gaussian function (equation 2.45) with σ cen = 0.3 ◦ , σ sur = 1.5 ◦ , and B = 5. C) The space-time receptive field of a cat LGN X cell. Note that both the center and surround regions reverse sign as a function of τ and that the temporal evolution is slower for the surround than for the center. D) A fit of the space-time receptive field in C using 2.46 with the same parameters for the Gaussian functions as in B, and temporal factors given by equation 2.47 with 1 /α cen = 16 ms for the center, 1/α sur = 32 ms for the surround, and 1 /β cen = 1/β sur = 64 ms. (A and C adapted from DeAngelis et al., 1995.) The ±sign allows both ON-center (+) and OFF-center (−) cases to be rep- resented. Figure 2.25B shows a spatial receptive field formed from the difference of two Gaussians that approximates the receptive field structure in figure 2.25A. Figure 2.25C shows that the spatial structure of the receptive field reverses over time with, in this case, a central ON region reversing to an OFF region as τ increases. Similarly, the OFF surround region changes to an ON region with increasing τ, although the reversal and the onset are slower for the surround than for the central region. Because of the difference between the time course of the center and surround regions, the space-time receptive field is not separable, although the center and surround components Draft: December 17, 2000 Theoretical Neuroscience 36 Neural Encoding II: Reverse Correlation and Visual Receptive Fields are individually separable. The basic features of LGN neuron space-time receptive fields are captured by the mathematical caricature D (x, y,τ)=±  D cen t (τ) 2πσ 2 cen exp  − x 2 + y 2 2σ 2 cen  − BD sur t (τ) 2πσ 2 sur exp  − x 2 + y 2 2σ 2 sur  . (2.46) Separate functions of time multiply the center and surround, but they can both be described by the same functions using two sets of parameters, D cen, sur t (τ) = α 2 cen ,sur τ exp (−α cen ,sur τ)−β 2 cen ,sur τ exp (−β cen ,sur τ) . (2.47) The parameters α cen and α sur control the latency of the response in the center and surround regions respectively, and β cen and β sur affect the time of the reversal. This function has characteristics similar to the function 2.29, but the latency effect is less pronounced. Figure 2.25D shows the space- time receptive field of equation 2.46 with parameters chosen to match figure 2.25C. time (ms) firing rate (Hz) Figure 2.26: Comparison of predicted and measured firing rates for a cat LGN neuron responding to a video movie. The top panel is the rate predicted by inte- grating the product of the video image intensity and a linear filter obtained for this neuron from a spike-triggered average of a white-noise stimulus. The resulting linear prediction was rectified. The middle and lower panels are measured firing rates extracted from two different sets of trials. (From Dan et al., 1996.) Figure 2.26 shows the results of a direct test of a reverse correlation model of an LGN neuron. The kernel needed to describe a particular LGN cell was first extracted using a white-noise stimulus. This, together with a rec- tifying static nonlinearity, was used to predict the firing rate of the neuron Peter Dayan and L.F. Abbott Draft: December 17, 2000 2.7 Constructing V1 Receptive Fields 37 in response to a video movie. The top panel in figure 2.26 shows the resulting prediction while the middle and lower panels show the actual firing rates extracted from two different groups of trials. The correlation coefficient between the predicted and actual firing rates was 0.5, which was very close to the correlation coefficient between firing rates extracted from different groups of trials. This means that the error of the prediction was no worse than the variability of the neural response itself. 2.7 Constructing V1 Receptive Fields The models of visual receptive fields we have been discussing are purely descriptive, but they provide an important framework for studying how the circuits of the retina, LGN, and primary visual cortex generate neural responses. In an example of a more mechanistic model, Hubel and Wiesel (1962) showed how the oriented receptive fields of cortical neurons could be generated by summing the input from appropriately selected LGN neurons. Their construction, shown in figure 2.27A, consists of alternating Hubel-Wiesel simple cell modelrows of ON-center and OFF-center LGN cells providing convergent input to a cortical simple cell. The left side of figure 2.27A shows the spatial arrangement of LGN receptive fields that, when summed, form bands of ON and OFF regions resembling the receptive field of an oriented simple cell. This model accounts for the selectivity of a simple cell purely on the basis of feedforward input from the LGN. We leave the study of this model as an exercise for the reader. Other models, which we discuss in chapter 7, include the effects of recurrent intracortical connections as well. In a previous section, we showed how the properties of complex cell responses could be accounted for using a squaring static nonlinearity. While this provides a good description of complex cells, there is little indication that complex cells actually square their inputs. Models of complex cells can be constructed without introducing a squaring nonlinearity. One such example is another model proposed by Hubel and Wiesel (1962), which is depicted in figure 2.27B. Here the phase-invariant response of a com- Hubel-Wiesel complex cell modelplex cell is produced by summing together the responses of several simple cells with similar orientation and spatial frequency tuning, but different preferred spatial phases. In this model, the complex cell inherits its orientation and spatial frequency preference from the simple cells that drive it, but spatial phase selectivity is reduced because the outputs of simple cells with a variety of spatial phases selectivities are summed linearly. Analysis of this model is left as an exercise. While the model generates complex cell responses, there are indications that complex cells in primary visual cortex are not exclusively driven by simple cell input. An alternative model is considered in chapter 7. Draft: December 17, 2000 Theoretical Neuroscience 38 Neural Encoding II: Reverse Correlation and Visual Receptive Fields LGN receptive fields simple cell On OffOff complex cell AB simple cells Figure 2.27: A) The Hubel-Wiesel model of orientation selectivity. The spatial arrangement of the receptive fields of nine LGN neurons are shown, with a row of three ON-center fields flanked on either side by rows of three OFF-center fields. White areas denote ON fields and grey areas OFF fields. In the model, the converging LGN inputs are summed linearly by the simple cell. This arrangement produces a receptive field oriented in the vertical direction. B) The Hubel-Wiesel model of a complex cell. Inputs from a number of simple cells with similar orientation and spatial frequency preferences ( θ and k), but different spatial phase preferences ( φ 1 , φ 2 , φ 3 ,andφ 4 ), converge on a complex cell and are summed linearly. This produces a complex cell output that is selective for orientation and spatial frequency, but not for spatial phase. The figure shows four simple cells converging on a complex cell, but additional simple cells can be included to give a more complete coverage of spatial phase. 2.8 Chapter Summary We continued from chapter 1 our study of the ways that neurons encode information, focusing on reverse-correlation analysis, particularly as applied to neurons in the retina, visual thalamus (LGN), and primary visual cortex. We used the tools of systems identification, especially the linear filter, Wiener kernel, and static nonlinearity to build descriptive linear and nonlinear models of the transformation from dynamic stimuli to time-dependent firing rates. We discussed the complex logarithmic map governing the way that neighborhood relationships in the retina are trans- formed into cortex, Nyquist sampling in the retina, and Gabor functions as descriptive models of separable and nonseparable receptive fields. Models based on Gabor filters and static nonlinearities were shown to account for the basic response properties of simple and complex cells in primary visual cortex, including selectivity for orientation, spatial frequency and phase, velocity, and direction. Retinal ganglion cell and LGN responses were modeled using a difference-of-Gaussians kernel. We briefly described simple circuit models of simple and complex cells. Peter Dayan and L.F. Abbott Draft: December 17, 2000 2.9 Appendices 39 2.9 Appendices A) The Optimal Kernel Using equation 2.1 for the estimated firing rate, the expression 2.3 to be minimized is E = 1 T  T 0 dt  r 0 +  ∞ 0 dτ D(τ)s(t −τ)−r( t)  2 . (2.48) The minimum is obtained by setting the derivative of E with respect to functional derivativethe function D to zero. A quantity, such as E, that depends on a function, D in this case, is called a functional, and the derivative we need is a functional derivative. Finding the extrema of functionals is the subject of a branch of mathematics called the calculus of variations. A simple way to define a functional derivative is to introduce a small time interval t and evaluate all functions at integer multiples of t.Wedefine r i = r(it) , D k = D(kt), and s i−k = s((i −k)t).Ift is small enough, the integrals in equation 2.48 can be approximated by sums, and we can write E =  t T T/t  i=0  r 0 +t ∞  k=0 D k s i −k −r i  2 . (2.49) E is minimized by setting its derivative with respect to D j for all values of j to zero, ∂E ∂D j = 0 = 2 t T T/t  i=0  r 0 + t ∞  k=0 D k s i−k − r i  s i−j  t. (2.50) Rearranging and simplifying this expression gives the condition t ∞  k=0 D k  t T T/t  i=0 s i−k s i−j  =  t T T/t  i=0 ( r i −r 0 ) s i−j . (2.51) If we take the limit t → 0 and make the replacements it → t, jt → τ , and kt → τ  , the sums in equation 2.51 turn back into integrals, the indexed variables become functions, and we find  ∞ 0 dτ  D(τ  )  1 T  T 0 dts(t −τ  )s(t −τ)  = 1 T  T 0 dt ( r(t) −r 0 ) s(t −τ). (2.52) The term proportional to r 0 on the right side of this equation can be dropped because the time integral of s is zero. The remaining term is the firing rate-stimulus correlation function evaluated at −τ, Q rs (−τ).The term in large parentheses on the left side of 2.52 is the stimulus autocorre- lation function. By shifting the integration variable t → t+τ,wefind that it is Q ss (τ −τ  ), so 2.52 can be re-expressed in the form of equation 2.4. Draft: December 17, 2000 Theoretical Neuroscience 40 Neural Encoding II: Reverse Correlation and Visual Receptive Fields Equation 2.6 provides the solution to equation 2.4 only for a white noise stimulus. For an arbitrary stimulus, equation 2.4 can be solved easily by the method of Fourier transforms if we ignore causality and allow the estimated rate at time t to depend on the stimulus at times later than t,so that r est (t ) = r 0 +  ∞ −∞ dτ D(τ)s(t −τ) . (2.53) The estimate written in this acausal form, satisfies a slightly modified ver- sion of equation 2.4,  ∞ −∞ dτ  Q ss (τ −τ  )D(τ  ) = Q rs (−τ) . (2.54) We define the Fourier transforms (see the Mathematical Appendix) ˜ D (ω ) =  ∞ −∞ dt D (t) exp (iω t) and ˜ Q ss (ω ) =  ∞ −∞ d τ Q ss (τ) exp( iωτ) (2.55) as well as ˜ Q rs (ω ) defined analogously to ˜ Q ss (ω ) . Equation 2.54 is solved by taking the Fourier transform of both sides and using the convolution identity (Mathematical Appendix)  ∞ −∞ dt exp(iωt)  ∞ −∞ dτ  Q ss (τ −τ  )D(τ  ) = ˜ D(ω ) ˜ Q ss (ω ) (2.56) In terms of the Fourier transforms, equation 2.54 then becomes ˜ D(ω ) ˜ Q ss (ω ) = ˜ Q rs (−ω) (2.57) which can be solved directly to obtain ˜ D(ω ) = ˜ Q rs (−ω)/ ˜ Q ss (ω ). The in- verse Fourier transform from which D (τ) is recovered is (Mathematical Appendix) D (τ) = 1 2π  ∞ −∞ dω ˜ D(ω ) exp(−iωτ) , (2.58) so the optimal acausal kernel when the stimulus is temporally correlated is given by D (τ) = 1 2π  ∞ −∞ dω ˜ Q rs (−ω) ˜ Q ss (ω ) exp(−iωτ) . (2.59) B) The Most Effective Stimulus We seek the stimulus that produces the maximum predicted responses at time t subject to the fixed energy constraint  T 0 dt   s (t  )  2 = constant . (2.60) Peter Dayan and L.F. Abbott Draft: December 17, 2000 2.9 Appendices 41 We impose this constraint by the method of Lagrange multipliers (see the Mathematical Appendix), which means that we must find the uncon- strained maximum value with respect to s of r est (t) +λ  T 0 dt  s 2 (t  ) = r 0 +  ∞ 0 dτ D(τ)s(t −τ)+λ  T 0 dt   s (t  )  2 (2.61) where λ is the Lagrange multiplier. Setting the derivative of this expression with respect to the function s to zero (using the same methods used in appendix A) gives D (τ) =−2 λs( t −τ) . (2.62) The value of λ (which is less than zero) is determined by requiring that condition 2.60 is satisfied, but the precise value is not important for our purposes. The essential result is the proportionality between the optimal stimulus and D (τ). C) Bussgang’s Theorem Bussgang (1952 & 1975) proved that an estimate based on the optimal kernel for linear estimation can still be self-consistent (although not necessar- ily optimal) when nonlinearities are present. The self-consistency condition is that when the nonlinear estimate r est = r 0 + F(L(t)) is substituted into equation 2.6, the relationship between the linear kernel and the firing rate-stimulus correlation function should still hold. In other words, we require that D (τ) = 1 σ 2 s T  T 0 dtr est (t)s(τ −t) = 1 σ 2 s T  T 0 dtF(L( t))s(τ −t). (2.63) We have dropped the r 0 term because the time integral of s is zero. In general, equation 2.63 does not hold, but if the stimulus used to extract D is Gaussian white noise, equation 2.63 reduces to a simple normalization condition on the function F. This result is based on the identity, valid for a Gaussian white-noise stimulus, 1 σ 2 s T  T 0 dtF(L( t))s(τ −t) = D(τ) T  T 0 dt dF (L(t)) dL . (2.64) For the right side of this equation to be D (τ), the remaining expression, involving the integral of the derivative of F, must be equal to one. This can be achieved by appropriate scaling of F. The critical identity 2.64 is based on integration by parts for a Gaussian weighted integral. A simplified proof is left as an exercise. Draft: December 17, 2000 Theoretical Neuroscience 42 Neural Encoding II: Reverse Correlation and Visual Receptive Fields 2.10 Annotated Bibliography Marmarelis & Marmarelis (1978), Rieke et al. (1997) and Gabbiani & Koch (1998) provide general discussions of reverse correlation methods. A useful reference relevant to our presentation of their application to the visual system is Carandini et al. (1996). Volterra and Wiener functional expansions are discussed in Wiener (1958) and Marmarelis & Marmarelis (1978). General introductions to the visual system include Hubel & Wiesel (1962, 1977), Orban (1984), Hubel (1988), Wandell (1995), and De Valois & De Valois (1990). Our treatment follows Dowling (1987) on processing in the retina, and Schwartz (1977), Van Essen et al. (1984), and Rovamo & Virsu (1984) on aspects of the retinotopic map from the eye to the brain. Prop- erties of this map are used to account for aspects of visual hallucinations in Ermentrout & Cowan (1979). We also follow Movshon et al. (1978a & b) for definitions of simple and complex cells; Daugman (1985) and Jones & Palmer (1987b) on the use of Gabor functions (Gabor, 1946) to describe visual receptive fields; and DeAngelis et al. (1995) on space-time receptive fields. Our description of the energy model of complex cells is based on Adelson & Bergen (1985), which is related to work by Pollen & Ronner (1982), Van Santen & Sperling (1984), and Watson & Ahumada (1985), and to earlier ideas of Reichardt (1961) and Barlow & Levick (1965). Heeger’s (1992; 1993) model of contrast saturation is reviewed in Carandini et al. (1996) and has been applied in a approach more closely related to the representational learning models of chapter 10 by Simoncelli & Schwartz (1999). The difference-of-Gaussians model for retinal and LGN receptive fields is due to Rodieck (1965) and Enroth-Cugell and Robson (1966). A useful reference to modeling of the early visual system is W ¨ org ¨ otter & Koch (1991). The issue of linearity and non-linearity in early visual processing is reviewed by Ferster (1994). Peter Dayan and L.F. Abbott Draft: December 17, 2000 Chapter 3 Neural Decoding 3.1 Encoding and Decoding In chapters 1 and 2, we considered the problem of predicting neural responses to known stimuli. The nervous system faces the reverse problem, determining what is going on in the real world from neuronal spiking pat- terns. It is interesting to attempt such computations ourselves, using the responses of one or more neurons to identify a particular stimulus or to extract the value of a stimulus parameter. We will assess the accuracy with which this can be done primarily by using optimal decoding techniques, regardless of whether the computations involved seem biologically plausi- ble. Some biophysically realistic implementations are discussed in chapter 7. Optimal decoding allows us to determine limits on the accuracy and re- liability of neuronal encoding. In addition, it is useful for estimating the information content of neuronal spike trains, an issue addressed in chapter 4. As we discuss in chapter 1, neural responses, even to a single repeated stimulus, are typically described by stochastic models due to their inher- ent variability. In addition, the stimuli themselves are often described stochastically. For example, the stimuli used in an experiment might be drawn randomly from a specified probability distribution. Natural stimuli can also be modeled stochastically as a way of capturing the statistical properties of complex environments. Given this two-fold stochastic model, encoding and decoding are related through a basic identity of probability theory called Bayes theorem. Let r represent the response of a neuron or a population of neurons to a stimulus characterized by a parameter s. Throughout this chapter, r = (r 1 , r 2 , ,r N ) for N neurons is a list of spike-count firing rates, although, for the present discussion, it could be any other set of parameters describing the neuronal response. Several different probabilities and con- Draft: December 17, 2000 Theoretical Neuroscience 2 Neural Decoding ditional probabilities enter into our discussion. A conditional probabilityconditional probability is just an ordinary probability of an event occurring except that its occur- rence is subject to an additional condition. The conditional probability of an event A occurring subject to the condition B is denoted by P[A |B]. The probabilities we need are: • P[s], the probability of stimulus s being presented. This is often called the prior probability,prior probability • P[r], the probability of response r being recorded, • P[r , s], the probability of stimulus s being presented and response r being recorded, • P[r |s], the conditional probability of evoking response r given that stimulus s was presented, and • P[s |r], the conditional probability that stimulus s was presented given that the response r was recorded. Note that P[r |s] is the probability of observing the rates r given that the stimulus took the value s, while P[r] is the probability of the rates taking the values r independent of what stimulus was used. P[r] can be computed from P[r |s] by summing over all stimulus values weighted by their probabilities, P[r] =  s P[r|s]P[s] and similarly P[s] =  r P[s|r]P[r] . (3.1) An additional relationship between the probabilities listed above can be derived by noticing that P[r , s] can be expressed as either the conditional probability P[r |s] times the probability of the stimulus, or as P[s|r] times the probability of the response, P[r , s] = P[r|s]P[s] = P[s|r]P[r]. (3.2) This is the basis of Bayes theorem relating P[s |r]toP[r|s],Bayes theorem P[s |r] = P[r|s]P[s] P[r] , (3.3) assuming that P[r] = 0. Encoding is characterized by the set of probabilities P[r |s] for all stimuli and responses. Decoding a response, on the other hand, amounts to determining the probabilities P[s |r]. According to Bayes theorem, P[s |r] can be obtained from P[r|s], but the stimulus probability P[s] is also needed. As a result, decoding requires knowledge of the statistical properties of experimentally or naturally occurring stimuli. In the above discussion, we have assumed that both the stimulus and response are characterized by discrete values so that ordinary probabilities, Peter Dayan and L.F. Abbott Draft: December 17, 2000 [...]... f a (s ) is approximately independent of s The roughly flat line in figure 3. 8 is proportional to this sum The constancy of the sum over tuning curves will be useful in the following analysis 1.0 f/rmax 0.8 0.6 0.4 0.2 0.0 -5 -4 -3 -2 -1 0 s 1 2 3 4 5 Figure 3. 8: An array of Gaussian tuning curves spanning stimulus values from -5 to 5 The peak values of the tuning curves fall on the integer values of... (3. 34) a=1 Maximizing this determines the MAP estimate, N T a=1 ra f a (sMAP ) p [sMAP ] = 0 + f a (sMAP ) p[sMAP ] (3. 35) If the stimulus or prior distribution is itself Gaussian with mean sprior and variance σprior , and we use the Gaussian array of tuning curves, equation 3. 35 yields sMAP = Draft: December 17, 2000 2 2 ra sa /σa + sprior /σprior T T 2 2 ra /σa + 1/σprior (3. 36) Theoretical Neuroscience. .. equation 3. 30 Setting the derivative to zero, we find that sML is determined by N ra a=1 f a (sML ) =0 f a (sML ) (3. 31) where the prime denotes a derivative If the tuning curves are the Gaussians of equation 3. 27, this equation can be solved explicitly using the re2 sult f a (s )/ f a (s ) = (sa − s )/σa , sML = Peter Dayan and L.F Abbott 2 ra sa /σa 2 ra /σa (3. 32) Draft: December 17, 2000 3. 3 Population... curve at sa = −5, −4, , 5 are the mean activities of the cells with preferred values at those locations for a stimulus at s = 0 1.0 0.5 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 sa Figure 3. 9: Simulated responses of 11 neurons with the Gaussian tuning curves shown in figure 3. 8 to a stimulus value of zero Firing rates for a single trial, generated using the Poisson model, are plotted as a function of the preferred... a (s ) f a (s ) (3. 43) If we assume that the array of tuning curves is symmetric, like the GausPeter Dayan and L.F Abbott Draft: December 17, 2000 3. 3 Population Decoding 25 1.0 f / rmax 0.8 IF / ( rmax T ) 0.6 0.4 0.2 0.0 -4 -3 -2 -1 0 1 2 3 4 s Figure 3. 11: The Fisher information for a single neuron with a Gaussian tuning curve with s = 0 and σa = 1, and Poisson variability The Fisher information... 2 0 -1 80 -9 0 0 90 180 0 -1 80 -9 0 0 90 180 wind direction (degrees) wind direction (degrees) Figure 3. 7: Maximum likelihood and Bayesian estimation errors for the cricket cercal system ML and Bayesian estimates of the wind direction were compared with the actual stimulus value for a large number of simulated firing rates Firing rates were generated as for figure 3. 5B The error shown is the root-mean-squared... (the Gaussian array in figure 3. 8, for example) and Poisson statistics can be computed from the conditional firing-rate probability in equation 3. 30 Because the spike-count rate is described here by a probability rather than a probability density, we use the discrete analog of equation 3. 41, IF ( s ) = − N d2 ln P[r|s] =T ra ds2 a=1 f a (s ) f a (s ) 2 − f a (s ) f a (s ) (3. 43) If we assume that the array... in figure 3. 3 Examination of figure 3. 3 suggests a relationship between the area under the ROC curve and the level of performance on the task When the ROC curve in figure 3. 3 lies along the diagonal, the area underneath it is 1/2, which is the probability of a correct answer in this case (given any threshold) When the task is easy and the ROC curve hugs the left axis and upper limit in figure 3. 3, and the... all the tuning curves have the same width, this reduces to sML = ra sa , ra (3. 33) which is a simple estimation formula with an intuitive interpretation as the firing-rate weighted average of the preferred values of the encoding neurons The numerator of this expression is reminiscent of the population vector Although equation 3. 33 gives the ML estimate for a population of neurons with Poisson variability,... vector v can be reconstructed from its Cartesian components through the component-weighted vector sum v = vx x +v y y Because the firing rates of Draft: December 17, 2000 Theoretical Neuroscience dot product 14 Neural Decoding A B c1 r1 8 error (degrees) c4 v r2 6 4 2 0 -1 80 c2 c3 -9 0 0 90 180 wind direction (degrees) Figure 3. 5: A) Preferred directions of four cercal interneurons in relation to the cricket’s . Fields in the Retina and LGN 35 03 0 200 0 0 3 3 -2 -1 0 1 2 0 100 200 0 D xt -2 -1 0 1 2 -2 0 2 0 D s B D A C Figure 2.25: Receptive fields of LGN neurons. A) The center-surround spatial structure. shown in figure 3. 3. Examination of figure 3. 3 suggests a relationship between the area under the ROC curve and the level of performance on the task. When the ROC curve in figure 3. 3 lies along the. equations 3. 7 and 3. 8, we can write β(z) =  ∞ z dr p[r|+ ] d β dz =−p[z|+ ] . (3. 13) Combining this result with 3. 8, we find that d β d α = dβ dz dz d α = p[z |+] p[z |−] = l(z), (3. 14) so the

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