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**by Michael Anthony Repucci**

Linear and Nonlinear Dynamics of Receptive Fields in Primary Visual Cortex (WELCOME EVERYONE) In this talk, I will highlight part of my thesis work on the linear and nonlinear dynamics of receptive fields in primary visual cortex. First, I will provide an introduction and describe some of the methods we used. Then, I will present the primary results, dealing firstly with the linear dynamics, secondly with the nonlinear dynamics, and thirdly with dynamic models. Finally, I will summarize the major findings and suggest how current models of the primary visual cortex could be improved. (NEXT) A thesis presentation at Weill Graduate School of Medical Science of Cornell University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Michael Anthony Repucci

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**The Visual Pathway Parvo Magno Left Eye Right Eye L R**

Shown here is a schematic of the visual pathway. (NEXT) Light is converted to an electrochemical signal by the photoreceptors. Neuronal signals pass through several layers of processing before reaching the ganglion cells of the retina, which send most of their axons via the optic nerve to the lateral geniculate nucleus of the thalamus, or LGN. (NEXT) In the LGN, which is often referred to as a relay nucleus, the passage of visual signals is modulated by diverse connections from many brain regions, and projections are sent, via the optic radiation, to the primary visual cortex, or V1. (NEXT) Visual processing continues in V1, as neuronal signals pass through several anatomically and functionally distinct layers. (NEXT) Of course, V1 is just the tip of the iceberg, shown here at the bottom of this chart. (PAUSE) V1 has been studied for more than 40 years, and is the first stage of cortical processing of visual information. Continued interest in V1 has been fueled by suggestions that V1 neurons may participate in many aspects of visual perception, including contour integration, visual texture segmentation, depth relations, and motion processing. My research focused on the properties of neurons in V1, which I will review presently. (NEXT)

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**The Experimental Tools**

Visual Stimulus Extracellular Electrode record V1 spikes (action potentials) We view the visual system (NEXT) as a black-box, and attempt to characterize it by describing how the visual stimulus is related to its response—that is, we examine the (NEXT) stimulus-response relationship. In this research, (NEXT) we presented several different types of visual stimuli, which I will describe shortly, to anaesthetized, paralyzed cats or monkeys, (NEXT) while recording the electrical signals, known as action potentials, or spikes, of V1 neurons using an extracellular electrode. (NEXT) ? s r stimulus -response relationship

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**Linear versus Nonlinear**

Linear Superposition rA+rB rB rA Nonlinear Suppression Nonlinear Facilitation sA sB sA+sB rB Before I describe the basic properties of V1 neurons, I would like to explain the difference between linear and nonlinear, as it relates to the stimulus-response relationship. (NEXT) A fundamental property of a linear system is superposition. If a V1 neuron was linear, (NEXT) then stimulus A would give response A, stimulus B would give response B, and the response to stimulus A plus stimulus B would be equal to response A plus response B. This is, in general, not true for a nonlinear system. As an example, (NEXT) consider this nonlinear system, in which stimulus A drives response A, and stimulus B drives response B, but stimulus A plus stimulus B leads to a response that is nonlinearly facilitated—that is, its value is greater than response A plus response B. On the other hand, (NEXT) in another nonlinear system, we might find that stimulus A drives response A, and stimulus B drives response B, but stimulus A plus stimulus B leads to a response that is nonlinearly suppressed—its value is less than response A plus response B. This distinction between linear and nonlinear systems (NEXT) also holds true for a time-varying, or dynamic, stimulus-response relationship. Although neurons are fundamentally nonlinear in nature, they can sometimes be reasonably approximated by a linear system. We asked whether this was true of V1 neurons by examining the linear and nonlinear components of the neuronal response. (NEXT) Holds true for dynamic relationship s(t) r(t) rA rB rA sA sB sA+sB

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**Retinal Receptive Fields**

Light Visual Space Light RESPONSE Light Light RGC off-cell on-cell Photoreceptors s-cone rod m-cone l-cone Receptive Field The stimulus-response characterization of a neuron is described by its receptive field. The concept of the receptive field is of central importance in visual perception, and in sensory physiology in general. (NEXT) In vision, the receptive field is defined as the region of visual space in which light elicits a neuronal response. (NEXT) Photoreceptors have small point-like receptive fields, (NEXT) whereas inputs to retinal ganglion cells are antagonistically arranged, such that their receptive fields are sensitive to local changes in luminance. This center-surround receptive field organization serves to accentuate contrast in the visual scene. (NEXT)

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**LGN and V1 Receptive Fields**

LGN neurons off-cell on-cell V1 neuron ` simple cell V1 neuron Similarly to retinal ganglion cells, LGN neurons have receptive fields with center-surround organization. (NEXT) In contrast, receptive fields of V1 neurons demonstrate an elongated structure, which makes them sensitive to oriented changes in contrast. As originally proposed by Hubel and Wiesel, (NEXT) feedforward connections from LGN neurons whose receptive fields are spatially aligned, help to create these oriented receptive fields in V1 neurons. But, as I will show, the receptive fields of V1 neurons are much more complicated than depicted in this cartoon, and cannot be explained by their LGN inputs alone. (NEXT)

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**The Grating Stimulus contrast orientation spatial spatial frequency**

To examine the structure of receptive fields in V1 neurons, researchers often use gratings. Gratings can be varied by their (NEXT) contrast, (NEXT) orientation, (NEXT) spatial frequency, and (NEXT) spatial phase—that is, the relative position of the grating. In addition, (NEXT) the temporal frequency of a grating can be non-zero. As I will show next with a few examples, the responses of V1 neurons are dependent on each of these five parameters. (PAUSE) Furthermore, recognize that any visual image can be reconstructed by a linear superposition of the appropriate gratings. (NEXT) spatial frequency spatial phase temporal frequency

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**V1 Contrast Response Function**

0% 25% 50% 75% 100% 10 20 30 Firing Rate (spikes/second) In this example, from our data, you can see how the average spike activity of a V1 neuron changes as a function of the contrast of a grating stimulus—this is referred to as the contrast response function. This sigmoid-shaped curve is typical of V1 neurons, but the point of saturation, which occurs at about 75% contrast in this neuron, can vary widely. (NEXT)

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**V1 Orientation Tuning 0° 90° 180° 270° 360° 10 20 30**

10 20 30 Firing Rate (spikes/second) Here, you can see how the average spike activity of a V1 neuron changes as a function of the orientation of a grating stimulus—this is referred to as orientation tuning. The orientation at which a V1 neuron gives its maximal response is referred to as its preferred orientation. Notice that V1 neurons may not be equally sensitive to gratings that drift in opposite directions. Furthermore, in many V1 neurons, the sharpness or selectivity of orientation tuning is greater than expected from its LGN inputs alone, which suggests that orientation tuning is refined in V1. (NEXT)

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**V1 Spatial Frequency Tuning**

0.125 0.25 0.5 1 2 4 8 Firing Rate (spikes/second) 5 10 15 20 cycles/degree In this example, you can see how the average spike activity of a V1 neuron changes as a function of the spatial frequency of a grating stimulus—this is referred to as spatial frequency tuning. The spatial frequency at which a V1 neuron gives its maximal response is referred to as its preferred spatial frequency. (PAUSE) Now, given the fact that V1 neurons are tuned to the parameters of gratings, as I’ve shown, an interesting point can be made. At every location in visual space, there are V1 neurons whose receptive fields are tuned to any given orientation and a variable range of spatial frequencies. Therefore, if V1 neurons were linear, a complete reconstruction of the visual image could be obtained by the appropriate superposition of their responses. (NEXT)

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**V1 Spatial Nonlinearities**

Classical Receptive Field (CRF) ` simple cell ` Non-Classical Receptive Field (NCRF) However, many of the perceptual roles often ascribed to V1 neurons require nonlinear mechanisms. As I’ve demonstrated, a V1 neuron can be stimulated by the appropriate grating positioned over its receptive field. (NEXT) When the grating is restricted to the so-called classical receptive field, a robust spike response is observed. (NEXT) As you might have suspected, if the grating is restricted to the surrounding region of visual space, which is referred to as the non-classical receptive field, the response of the V1 neuron cannot be distinguished from its spontaneous spike activity. (NEXT) However, in many V1 neurons, concurrent stimulation of the classical and non-classical receptive fields produces a response that is not equal to sum of the responses to these two gratings individually. This is an example of one type of spatial nonlinearity found in the receptive fields of V1 neurons. As depicted here, when the gratings in the classical and non-classical receptive fields are both at the preferred orientation and spatial frequency, the nonlinear effect of the non-classical receptive field is typically suppressive—this is referred to as iso-oriented suppression. Because this effect is orientation-dependent, it must derive from mechanisms within the visual cortex. Researchers have speculated that the nonlinear influences of the non-classical receptive field may be involved in the perception of extended contours, or changes in the local visual texture. (NEXT) r(t) = CRF r(t) = NCRF r(t) = CRF + NCRF ≠ r(t) + r(t)

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**V1 Size Tuning 0° 10° 4 8 12 Firing Rate (spikes/second) 0° 4° 8° 10**

4 8 12 Firing Rate (spikes/second) 0° 4° 8° 10 20 30 In these examples, you can see how the average spike activity of two V1 neurons change as a function of the size of a grating stimulus—this is referred to as a size tuning, or area-summation, curve. The size tuning curve provides one estimate of the size of the classical receptive field, and assays for the strength of iso-oriented suppression from the non-classical receptive field. Notice that some V1 neurons exhibit marked suppression as the grating size increases, while others do not. (NEXT)

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**The Goals s(t) r(t) V1 neuron Nonlinear? Linear and Nonlinear Dynamics**

NCRF CRF Nonlinear? s(t) r(t) Up until this point, I have not mentioned how the response of a V1 neuron changes in time—unfortunately, much vision research likewise ignores these receptive field dynamics. However, it is well known that the receptive fields of V1 neurons are dynamic. This stems in part from neuronal conduction delays, and the fact that the receptive fields of retinal ganglion cells and LGN neurons are also dynamic. But, as I will show, the receptive fields of some V1 neurons demonstrate additionally complex dynamics that presumably derive from cortical mechanisms. Therefore, my thesis work aimed to simultaneously characterize the (NEXT) linear and nonlinear dynamics of receptive fields in primary visual cortex in a spatially and temporally specific manner. (NEXT) Linear and Nonlinear Dynamics of Receptive Fields in Primary Visual Cortex

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**The Stimulus Tokens Classical Receptive Field (CRF) Non-Classical**

(NCRF) ` V1 neuron To do this, we designed a stimulus that uses a rapid, pseudo-random sequence of stationary gratings and blanks—what I will call the tokens—placed over and around the receptive field of a V1 neuron. (NEXT) The special mathematical properties of this token sequence, which are determined by a non-binary m-sequence, facilitate the analysis. This entire sequence, which lasts approximately five minutes, is presented four times to each neuron, so that error estimates can be obtained. As indicated, the center of the stimulus covers the classical receptive field, and the surround of the stimulus covers the non-classical receptive field. We used gratings that varied by orientation, as shown here, or by spatial frequency, or by spatial phase. Today, I will present only the results of experiments in which the orientation of the gratings was varied. (NEXT) Tokens

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**The Linear Dynamics Orientation Tuning (separable or “unimodal”)**

Firing Rate (spikes/second) Classical Receptive Field 17 22 27 7 12 Non-Classical Receptive Field 20 ms 40 60 80 100 120 To examine the linear dynamics of orientation tuning in V1 neurons, we correlate the spike activity of a V1 neuron with each of the tokens at several physiologically-relevant stimulus-response delays, (NEXT) separately in either the classical or non-classical receptive field. Notice that the horizontal grating in all figures indicates the relative preferred orientation for that neuron, and that the x-axis marks the discrete measurements we made of the token-dependent spike rate. (NEXT) By 20 milliseconds after the presentation of any token, there is no orientation tuning present in the response of this V1 neuron, in either the classical or non-classical receptive fields. (NEXT) By 40 milliseconds, however, this V1 neuron shows a robust response to its preferred orientation in the classical receptive field only. (NEXT) This response falls of slightly at 60 milliseconds, (NEXT) and is gone by 80, (NEXT) 100, or (NEXT) 120 milliseconds. This neuron is typical of 98% of our population, in that its peak response occurs between 40 and 60 milliseconds after stimulus presentation, and it shows a preference for the same orientation at all times—(NEXT) this is referred to as orientation-time separable or unimodal dynamics. The fact that there is no linear response in the non-classical receptive field is consistent with a modulatory, rather than linear additive, role for non-classical receptive field influences. These observations agree in large part with previous reports in the literature. (NEXT)

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**Firing Rate (spikes/second)**

The Linear Dynamics Orientation Tuning (“multimodal”) 8 12 16 Firing Rate (spikes/second) CRF 20 ms 40 60 80 100 120 6 12 18 CRF 10 20 CRF Multimodal dynamics were observed in 11% of V1 neurons. (NEXT) This includes neurons that show inversions in orientation preference, (NEXT) inseparable shifts in orientation preference, (NEXT) and “Mexican-hat” shaped orientation tuning, in which orientations flanking the preferred orientation evoke a smaller response than the other non-preferred orientations, as highlighted by the arrows. Notice that these types of dynamics cannot be explained by their feedforward LGN inputs alone, and therefore support the suggestion that orientation tuning is refined by cortical processing. Although they are rare, V1 neurons like those in the left and middle panels may play important roles in signaling rapid changes in the local orientation content of the visual image, while V1 neurons like that in the right panel provide clues into the nature of the sharpening of orientation tuning, which is believed to arise in part from cortical suppression at orientations flanking the preferred orientation. (NEXT)

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**The Analysis Stimulus s(t) r(t) Token-dependent spike rate Spikes**

Time (milliseconds) 20 40 60 160 180 200 80 100 120 140 Stimulus s(t) pairs across time (e.g., tokens in the classical receptive field at 40 and 60 milliseconds prior) pairs across space and time (e.g, tokens in the classical receptive field and the non-classical receptive field) single tokens 60 milliseconds prior single tokens 40 milliseconds prior single tokens 20 milliseconds prior Token-dependent spike rate correlate spikes with the occurrence of single tokens or pairs of tokens at various stimulus-response delays (kernel) Before I show the results of the nonlinear dynamics, I would like to explain more carefully how the stimulus-response characterization is performed. The analysis, which is often referred to as reverse correlation or the spike-triggered average, involves estimating the (NEXT) token-dependent spike rate by (NEXT) correlating spikes with the occurrence of single tokens or pairs of tokens at various, physiologically-relevant stimulus-response delays. The collection of estimates across all tokens or pairs of tokens is called a kernel. As I showed, first-order kernels are calculated at a (NEXT) 20 millisecond stimulus-response delay, (NEXT) 40 millisecond, (NEXT) 60 millisecond, et cetera, and describe the linear dynamics of orientation tuning in V1 neurons. Second-order kernels are are calculated at (NEXT) several pairs of post-stimulus delays in the classical receptive field, or (NEXT) across pairs of receptive field regions, and describe the nonlinear dynamics of orientation tuning in V1 neurons. (NEXT) Second-order kernels indicate structure in the response of V1 neurons that is not accounted for by linearity. (NEXT) second-order kernels (nonlinear dynamics) reflect response structure not accounted for by the linear dynamics

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**The Nonlinear Dynamics**

Orientation Tuning (CRF and NCRF) rarely significant at all physiologically-relevant stimulus-response delays relatively weak even in neurons that showed significant iso-oriented suppression on size tuning curves 20 40 60 160 180 200 80 100 120 140 To our surprise, the nonlinear dynamics of orientation tuning between the classical and non-classical receptive fields, at all physiologically-relevant stimulus-response delays, (NEXT) were rarely significant and (NEXT) relatively weak, even in V1 neurons that showed significant iso-oriented suppression. Unfortunately, I do not have time to discuss why this is true. (NEXT)

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**The Nonlinear Dynamics**

Orientation Tuning (CRF) Dark Red points have a mean that is significantly greater than zero, signifying nonlinear facilitation 5 10 15 20 25 30 Firing Rate (spikes/second) Dark blue points have a mean that is significantly less than zero, signifying nonlinear suppression First Token Pink points have a mean greater than zero but are not significant Light blue points have a mean less than zero but are not significant Instead, (NEXT) I will focus on the nonlinear dynamics of orientation tuning within the classical receptive field, which showed significant and consistent nonlinearities in more than half of the V1 neurons in our population. For the majority of V1 neurons, the strongest nonlinear interactions occurred between tokens at a (NEXT) 60 or 80 millisecond stimulus-response delay followed immediately by tokens at a (NEXT) 40 or 60 millisecond stimulus-response delay. Notice that I have partitioned the lower-left portion of the figure to accentuate the interaction between gratings and the blank token. If a V1 neuron were completely linear, the predicted kernel values would be zero at each point; we used a conservative measure to gauge significance. (NEXT) Red points are significantly greater than zero, corresponding to nonlinear facilitation. (NEXT) Blue points are significantly less than zero, corresponding to nonlinear suppression. (NEXT) Paler points have the same sign but are not significant. (NEXT) A scale bar is added to indicate the influence on the spike rate. Points indicating an interaction between the preferred orientation, as obtained from the analysis of the linear dynamics, and any other token are circled in doubly-thick lines. There are three nonlinear interactions I would like to draw your attention to. (NEXT) First a facilitation for the preferred grating followed by the preferred grating. This indicates a special sensitivity to an extended presentation of the preferred grating. (NEXT) Second an asymmetry between facilitation for the blank followed by the preferred grating, paired with a suppression for the preferred grating followed by the blank. This suggests the presence of an orientation-specific contrast gain control mechanism. Contrast gain control is a wide-spread mechanism used by neurons in the visual system, that serves to reduce the response of a neuron during a period of high contrast. (EXPLAIN?) The nonlinearities seen in this V1 neuron are representative of about half of the V1 neurons. (NEXT) Second Token

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**The Nonlinear Dynamics**

Orientation Tuning (CRF) 1 2 Firing Rate (spikes/second) First Token The nonlinearities observed in the other half of V1 neurons are represented by this example. Here, we can see the same relative asymmetry between facilitation for the blank followed by the preferred grating, and suppression for the preferred grating follow by the blank—although in fact the latter interaction is not significant. In contrast to the previous example I showed, an extended presentation of the preferred grating leads to nonlinear suppression. (NEXT) Second Token

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**The Nonlinear Dynamics**

Population Summary Cat Monkey Preferred-Preferred Grating Asymmetry Shown here is the distribution for these values across the V1 neurons in our population. The position along the x-axis corresponds to the relative strength of the nonlinearity for the preferred grating followed by the preferred grating, where points to the left of the dashed line are suppressive and points to the right are facilitative. The position along the y-axis corresponds to the relative strength of the nonlinearity for the blank followed by the preferred grating minus the nonlinearity for the preferred grating followed by the blank, where points above the dashed line indicate a relative facilitation for blank followed by preferred and suppression for preferred followed by blank, as was shown in the previous two examples, and points below the dashed line indicate a relative suppression for blank followed by preferred and facilitation for preferred followed by blank, which runs counter to the proposed contrast gain control mechanism. As I showed in the previous two examples, most V1 neurons show an asymmetry consistent with an orientation specific contrast gain control mechanism. Furthermore, the population is about equally divided between nonlinear facilitation or suppression for an extended presentation of the preferred grating. This suggests that some V1 neurons may be particularly sensitive to stimulus constancy. (NEXT)

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**? Dynamic Models Static Nonlinearity Models (LNP) s(t) r(t)**

linear component (L) fit to empirical linear dynamics static nonlinearity (N) threshold (prevent negative spike rate) square-root linear squared probabilistic spike generation (P) Finally, to see if the observed nonlinearities could be explained by dynamic models, (NEXT) we created three static nonlinearity models of each V1 neuron. A static nonlinearity is one that is not time dependent. Static nonlinearity models, or LNP models, are used commonly in neuroscience, even though they are overly simplistic, as a first attempt to explain empirical observations. (NEXT) The linear component of our LNP models were fit as closely as possible to the linear dynamics observed in each V1 neuron. (NEXT) Static nonlinearities were added that included (NEXT) a spike threshold—to prevent negative spike rates—followed by one of three power-laws chosen to be consistent with those used by other researchers, including (NEXT) a square-root, (NEXT) a linear, or (NEXT) a squared power. Finally, the model uses a probabilistic spike generating mechanism whose statistics are Poisson, to simulate the spike activity of V1 neurons. (NEXT) We feed each model the same stimulus sequence, and use its response to calculate the linear and nonlinear dynamics of the model, which we compare to the dynamics observed in V1 neurons. (NEXT) ? L(τ) s(t) = r(t)

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**Firing Rate after Nonlinearity**

Dynamic Models Effect of the Static Nonlinearity Firing Rate before Nonlinearity (spikes/second) Firing Rate after Nonlinearity (spikes/second) 10 20 30 40 Threshold-Linear Threshold-Squared Threshold-Square-Root Show here is an example, from the model fits to a single V1 neuron, of the relationship between the predicted spike rate before adding the nonlinearity—that is, for the linear half of the model on the x-axis—and the predicted spike rate after the nonlinearity on the y-axis. As you can see, the threshold has prevented any negative spike rates. (NEXT)

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**Dynamic Models Linear Dynamics Firing Rate (spikes/second) V1 Neuron**

17 22 27 7 12 20 ms 40 60 80 100 120 Model: Threshold-Linear As can been seen in this example, the linear dynamics of the threshold-linear model, shown on the right, closely match those obtained from the V1 neuron, shown on the left. (NEXT)

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**Strength of Linear Dynamics**

Dynamic Models Population Summary V1 Neurons Models Strength of Linear Dynamics Threshold-Square-Root Threshold-Squared Threshold-Linear 10 20 This is generally true for the models of V1 neurons. When we compare the strength of the linear dynamics of V1 neurons, on the x-axis, to that obtained from the models, on the y-axis, we see that, across the population, all three models generally provide a good fit to the empirical data—a perfect fit would fall on the dashed line. This indicates that, as expected, the models were able to reproduce the linear component of the responses of V1 neurons, despite the added static nonlinearities. (NEXT)

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**Strength of Nonlinear Dynamics**

Dynamic Models Population Summary V1 Neurons Models Strength of Nonlinear Dynamics Threshold-Square-Root Threshold-Squared Threshold-Linear 4 8 12 In contrast, the strength of the nonlinear dynamics of V1 neurons were not well accounted for by the static nonlinearity models. Points below the dashed line indicate that the nonlinear dynamics of a V1 neuron were stronger than those produced by its models—this was true in all but one V1 neuron. Notice that all of the models produce non-zero valued nonlinear dynamics, indicating the presence in the model system of a nonlinearity. However, (NEXT) a relatively small fraction of the models are deemed significant by the same criterion used to judge the nonlinear dynamics of V1 neurons, as indicated by those points that fall above the horizontal dashed line. (NEXT)

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**Dynamic Models Nonlinear Dynamics 10 20 30 Firing Rate (spikes/second)**

10 20 30 Firing Rate (spikes/second) Second Token First Token V1 Neuron Model: Threshold-Square Root Model: Threshold-Squared Model: Threshold-Linear Nevertheless, we would like to see whether the nonlinear dynamics produced by the models show qualitatively similar features. On the left is shown the nonlinear dynamics of one of the V1 neurons I presented earlier. (NEXT) On the right are the nonlinearities produced by the threshold-square-root model for that neuron. As you can see from the scale bars, the largest nonlinearity produced in the threshold-square-root model is about one-third of the magnitude of that observed in the V1 neuron. Furthermore, where the V1 neuron showed a nonlinear facilitation for the preferred grating followed by the preferred grating, this model shows nonlinear suppression. This results from the suppressive, or saturating, nature of the threshold-square-root model. Furthermore, the asymmetry between facilitation for the blank followed by the preferred grating and suppression for the preferred grating followed by the blank is not significant in this model. Similarly, (NEXT) the threshold-linear model does not reproduce the nonlinear dynamics of the V1 neuron either in magnitude or direction. However, (NEXT) the threshold-squared model does show nonlinear facilitation in response to an extended presentation of the preferred grating and suppression for the preferred grating followed by the blank, even though the magnitude is less than half as large. This suggests that a squaring, or accelerating, nonlinearity may be at least partly responsible for the nonlinear dynamics observed in this V1 neuron. (NEXT)

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**Dynamic Models Nonlinear Dynamics Asymmetry**

population summary Preferred-Preferred Grating Asymmetry Threshold-Square-Root Threshold-Squared Threshold-Linear Across the population, you can see that these simple static nonlinearity models do not do a good job in general at reproducing the observed nonlinear dynamics of V1 neurons. Recall that the analogous plot to this one for the V1 neurons showed that most neurons were in the two upper quadrants, indicating an asymmetry involving nonlinear facilitation for the blank followed by the preferred grating and nonlinear suppression for the preferred grating followed by the blank, coupled with a range of values from strong nonlinear suppression to strong nonlinear facilitation for an extended presentation of the preferred grating. Although threshold-square-root models usually result in nonlinear suppression in response to an extended presentation of the preferred grating, and threshold-squared models typically produce nonlinear facilitation, none of the static nonlinearity models account well for the asymmetry. Thus, the nonlinear mechanisms represented by these models are likely to account only for a small fraction of the nonlinear dynamics observed in V1 neurons. (NEXT)

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**Conclusions Linear Dynamics Nonlinear Dynamics Dynamic Models**

nearly all V1 neurons show separable (unimodal) dynamics multimodal dynamics may be important for signaling rapid changes in the local orientation content of the visual stimulus Nonlinear Dynamics CRF versus NCRF nonlinearities were weak or non-significant even in V1 neurons that showed strong iso-oriented suppression on size tuning curves CRF significant nonlinearities in more than half of V1 neurons nonlinear facilitation or suppression for presentation of the preferred grating followed by the preferred grating suggests that V1 neurons are sensitive to stimulus constancy asymmetry involving nonlinear facilitation for the blank followed by the preferred grating and suppression for the preferred grating followed by the blank suggests the presence of an orientation-specific contrast gain control mechanism Dynamic Models simple static nonlinearity models cannot explain the magnitude of the nonlinear dynamics of V1 neurons saturating (square-root) and accelerating (squared) nonlinear mechanisms may have contributed to the observed nonlinear dynamics in V1 neurons In conclusion, we created a novel stimulus to simultaneously characterize the linear and nonlinear dynamics of receptive fields of V1 neurons…. (NEXT)

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Ouch! The End Thank you. Are there any questions?

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Linear Dynamics Spatial Frequency Tuning (separable)

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Linear Dynamics Spatial Frequency Tuning (inseparable)

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Linear Dynamics Shift in Preferred Spatial Frequency

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Nonlinear Dynamics Spatial Frequency – CRF

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Nonlinear Dynamics Spatial Frequency – CRF

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**Linear Dynamics Spatial Phase Tuning (separable) B 90 180 270 -1.5 -1**

90 180 270 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 Spatial Phase (degrees) Firing Rate Modulation (spikes/second) Classical Receptive Field 20 ms 40 60 80 100 120 C3002T E11

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Linear Dynamics Spatial Phase Tuning (insensitive and inversion)

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Linear Dynamics Spatial Phase Tuning (inseparable)

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Linear Dynamics Phase sensitivity does not agree with simple/complex distinction (i.e., F1/F0 ratio)

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Nonlinear Dynamics Spatial Phase – CRF

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Nonlinear Dynamics Spatial Phase – CRF

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Nonlinear Dynamics Spatial Phase – CRF temporal development

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**Phase-Dependent Analysis**

First-Order Kernel 5 10 15 20 25 Tokens Firing Rate (spikes/second) Classical Receptive Field 20 msec 40 msec 60 msec 80 msec 100 msec 120 msec

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