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Linear and Nonlinear Dynamics of Receptive Fields in Primary Visual Cortex 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 Right Eye Left Eye Right Eye Left Eye L R Magno Parvo

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The Experimental Tools sr ? Visual Stimulus - r esponse s timulus relationship Extracellular Electrode record V1 spikes (action potentials)

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Linear versus Nonlinear Nonlinear Suppression Nonlinear Facilitation sAsA sBsB s A +s B rArA rBrB r A +r B sAsA sBsB s A +s B rArA rBrB rArA rBrB Linear Superposition Holds true for dynamic relationship s(t) r(t)

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Retinal Receptive Fields Visual Space Light RESPONSE Light Receptive Field Photoreceptors s-cone rod m-cone l-cone RGC off-cell on-cell

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LGN and V1 Receptive Fields LGN neurons off-cell on-cell V1 neuron ` simple cell

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The Grating Stimulus contrast spatial frequency spatial phase orientation temporal frequency

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V1 Contrast Response Function 0%25%50%75%100% 10 20 30 Firing Rate (spikes/second)

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V1 Orientation Tuning 0°0°90°180°270°360° 0 10 20 30 Firing Rate (spikes/second)

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V1 Spatial Frequency Tuning 0.1250.250.51248 Firing Rate (spikes/second) 5 10 15 20 cycles/degree

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V1 Spatial Nonlinearities ` simple cell Non-Classical Receptive Field (NCRF) Classical Receptive Field (CRF) ` r(t) =CRF r(t) =NCRF r(t) =CRF + NCRF≠ r(t) + r(t)

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V1 Size Tuning 0°10° 0 4 8 12 Firing Rate (spikes/second) 0°4°8° 0 10 20 30

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The Goals V1 neuron NCRF CRF Nonlinear? Linear and NonlinearDynamicsof Receptive Fields in Primary Visual Cortex s(t) r(t)

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Non-Classical Receptive Field (NCRF) Classical Receptive Field (CRF) The Stimulus Tokens ` V1 neuron

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Non-Classical Receptive Field Firing Rate (spikes/second) Classical Receptive Field 17 22 27 7 12 The Linear Dynamics Orientation Tuning 20 ms 40 60 80 100 120 (separable or “unimodal”)

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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 0 10 20 CRF

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–second-order kernels (nonlinear dynamics) reflect response structure not accounted for by the linear dynamics Token-dependent spike rate –correlate spikes with the occurrence of single tokens or pairs of tokens at various stimulus-response delays (kernel) Spikes r(t) Time (milliseconds) 204060160180200 80100 120140 Stimulus s(t) single tokens 20 milliseconds prior single tokens 40 milliseconds prior single tokens 60 milliseconds prior 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) The Analysis

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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 204060160180200 80100 120140

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Orientation Tuning (CRF) The Nonlinear Dynamics Second Token First Token 0 5 10 15 20 25 30 Firing Rate (spikes/second) Dark Red points have a mean that is significantly greater than zero, signifying nonlinear facilitation 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 Dark blue points have a mean that is significantly less than zero, signifying nonlinear suppression

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Orientation Tuning (CRF) The Nonlinear Dynamics 0 1 2 Firing Rate (spikes/second) Second Token First Token

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The Nonlinear Dynamics Cat Monkey Preferred-Preferred Grating Asymmetry Population Summary

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Dynamic Models s(t)r(t) ? = L(τ) Static Nonlinearity Models (LNP) –linear component (L) fit to empirical linear dynamics –static nonlinearity (N) threshold (prevent negative spike rate) –square-root –linear –squared –probabilistic spike generation (P)

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Dynamic Models Effect of the Static Nonlinearity Firing Rate before Nonlinearity (spikes/second) Firing Rate after Nonlinearity (spikes/second) 0102030 0 20 40 Threshold-Linear Threshold-Squared Threshold-Square-Root

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Dynamic Models Firing Rate (spikes/second) V1 Neuron 17 22 27 7 12 20 ms 40 60 80 100 120 Linear Dynamics Model: Threshold-Linear

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Dynamic Models Population Summary V1 Neurons Models Strength of Linear Dynamics Threshold-Square-Root Threshold-Squared Threshold-Linear 1020 10 0

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Dynamic Models Population Summary V1 Neurons Models Strength of Nonlinear Dynamics Threshold-Square-Root Threshold-Squared Threshold-Linear 04812 4 8

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0 10 20 30 Firing Rate (spikes/second) Second Token First Token V1 NeuronModel: Threshold-Square Root Dynamic Models Nonlinear Dynamics Model: Threshold-LinearModel: Threshold-Squared

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Dynamic Models Preferred-Preferred Grating Asymmetry Threshold-Square-Root Threshold-Squared Threshold-Linear Nonlinear Dynamics –population summary

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Linear Dynamics –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 Conclusions

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The End Ouch!

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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)

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Spatial Phase Tuning (insensitive and inversion) Linear Dynamics

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Spatial Phase Tuning (inseparable) Linear Dynamics

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

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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

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