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## Sensorimotor Transformations

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**Sensorimotor Transformations**Maurice J. Chacron and Kathleen E. Cullen**Outline**• Lecture 1: - Introduction to sensorimotor transformations - The case of “linear” sensorimotor transformations: refuge tracking in electric fish - introduction to linear systems identification techniques - Example of sensorimotor transformations: Vestibular processing, the vestibulo-occular reflex (VOR).**Outline**• Lecture 2: - Nonlinear sensorimotor transformations - Static nonlinearities - Dynamic nonlinearities**Lecture 1**Sensorimotor transformation: if we denote the sensory input as a vector S and the motor command as M, a sensorimotor transformation is a mapping from S to M : M =f(S) Where f is typically a nonlinear function**Examples of sensorimotor transformations**• Vestibulo-occular reflex • Reaching towards a visual target, etc…**Refuge tracking**Sensory input Motor output Error**Results**• Tracking performance is best • when the refuge moves slowly • Tracking performance degrades when • the refuge moves at higher speeds • There is a linear relationship between sensory input and motor output (Cowan and Fortune, 2007)**Linear functions**• What is a linear function? • So, a linear system must obey the following definition:**Linear functions (continued)**• This implies the following: a stimulus at frequency f1 can only cause a response at frequency f1**Linear transformations**assume output is a convolution of the input with a kernel T(t) with additive noise. We’ll also assume that all terms are zero mean. • Convolution is the most general linear • transformation that can be done to a signal**An example of linear coding:**• Rate modulated Poisson process time dependent firing rate time**Linear Coding:**Example: Recording from a P-type Electroreceptor afferent. There is a linear relationship between Input and output Gussin et al. 2007 J. Neurophysiol.**Fourier decomposition and transfer functions**- Fourier Theorem: Any “smooth” signal can be decomposed as a sum of sinewaves • Since we are dealing with linear transformations, • it is sufficient to understand the nature of linear • transformations for a sinewave**Linear transformations of a sinewave**• Scaling (i.e. multiplying by a non-zero constant) • Shifting in time (i.e. adding a phase)**Cross-Correlation Function**For stationary processes: In general,**Cross-Spectrum**• Fourier Transform of the Cross-correlation function • Complex number in general a: real part b: imaginary part**Representing the cross-spectrum:**: amplitude : phase**Transfer functions (Linear Systems Identification)**assume output is a convolution of the input with a kernel T(t) with additive noise. We’ll also assume that all terms are zero mean. Transfer function**Calculating the transfer function**and average over noise realizations multiply by: =0**Sinusoidal stimulationat different frequencies**Response Stimulus 20 msec**Combining transfer functions**input output**Vestibular system**Cullen and Sadeghi, 2008**Example: vestibular afferents**CV=0.044 CV=0.35**Regular afferent**120 ` 100 Firing rate (spk/s) 80 60 40 20 Head velocity (deg/s) 0 -20 -40**Irregular afferent**160 ` 140 120 100 80 Firing rate (spk/s) 60 40 20 Head velocity (deg/s) 0 -20 -40**Using transfer functions to characterize and model refuge**tracking in weakly electric fish Sensory input Motor output Error**Characterizing the sensorimotor transformation**1st order 2nd order**Modeling refuge tracking using transfer functions**sensory processing sensory input motor processing motor output**Modeling refuge tracking using transfer functions**sensory processing sensory input motor output Newton**Summary**• Some sensorimotor transformations can be described by linear systems identification techniques. • These techniques have limits (i.e. they do not take variability into account) on top of assuming linearity.