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by: Walker Witting


Marketplace > Rice University > Applied Mathematics > CAAM 415 > THEORETICAL NEUROSCIENCE
Walker Witting
Rice University
GPA 3.58


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Class Notes
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This 20 page Class Notes was uploaded by Walker Witting on Monday October 19, 2015. The Class Notes belongs to CAAM 415 at Rice University taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/225003/caam-415-rice-university in Applied Mathematics at Rice University.

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Date Created: 10/19/15
The neural basis of Bayesian inference Lemme CE Optimal cue integration flat prior Behavioral model 17S anxV 0C p xA xV S pxA SpxV S Neural model P5 FADrV 0C prArV l 5 prAlsprVS Response estimate distribution has Uncertainty is encoded through PPC E A i 1 ActIVIty O O O O 0 Probability I Preferred stimulus Question how is this PPC used to implement the computation psrArVocprAsprVis Neural variability model Assume only for now independent Poisson variability N 61300 S 439 prlsH i1 73 Lecture 1 9 can be rewritten as r i 01 hsr N pl lS e Wlth SZGXPE W his10gfis General POIssonllke variability 01 hsr prlsnse Fano can be unequal to 1 correlations allowed 9 more realistic Relationship between hs tuning curve and covariance 1 ehsr 77S f sltrgt 2sltrrTgt ltrgtltrTgt Exercise given prs Show that Implementation of optimal cue integration n 17SII AV psrAV OCpS1 ApS1 V Auditory input gt 20 E r A quot 2 10 39039 o rAVrAr V Preferred stimulus 3 39 Visual input 2 f lt1 20 39 4 1 v 0 Preferred stimulus g 10 aquot a Multisensory layer 0 0 5 Preferred stimulus If rA and rV are Poisson like with the same hs then addition implements optimal cue integration Third population FM 2 rA rV prAV l SHP1quotA l SPI V S51 AV I A I39VdrAdrV JJWexpww FA eXphS rV5rAV FA rVdrAdrV 2 W 77AS77VS PAV FAV nAltsgtnVltsgt eXphs rAVdrA exphs rAV prAV I 5 exphSOF1V 2 DA FAlt0V1 V 77AS77VS 0A FAlt0V1 V 77A 577VS prAISprVIS eXphS39rAeXphS39rI eXphS FAV Same dependence on 5 therefore prAV I S 0C prA I SprV I 5 pS I rAV 0C prA I SprV I S Relating back to behavior Lecture 1 Fisher information 12 2L2 L2 014V GA CV 1 62 lt gt sA sV 031V 20 0 estimate Fisher information for independent Poisson variability N l s fl S as Exercises Compute Fisher information for Poisson like variability Show that rAV rArVimplies 12 1 1 AV A 2 2 0V Special case identical tuning curves and covariance matrices Activity Preferred stimulus Probability ps 1 Bayes rule O Stimulus pS 1 Bayes rule a Stimulus Gain and precision 1 goCIS gainof 02 population variance of estimate distribution High gain high certainty not guaranteed Multisensory gain and precision Audition Vision 0 0 gV O O l Preferred stimulus 39 1 1 1 0quot 0 Thus 2 2 2 AV A 0V Preferred stimulus What if hs not the same in both populations Instead of rAV rA rV Linear combination rAV WArA WVrV WA and WV are synaptic weights fixed across trials 9 Very general scheme for optimal cue combination So far adding spike counts Does this work for realistic neurons Conductancebased integrateandfire neuron Example excitatory neuron Conductances Currents Membrane potl 50 100 150 200 250 300 350 400 450 500 Timems C amp TCt EL giErV1t EE g tVt EI gm Vit EA IntegrateAndFire Network Decoder Auditoryvisual layer conductancebased V V I7 integrateandfire neurons 0 Q a O Q g 0 Q Gaussian connectivity Visual input Auditory input Spiking neurons with nearPoisson statistics and Gaussian tuning curves Simulating cue combination Visual input only Obtain mean and variance of visual estimates from output spike counts Repeat for auditory input only Multisensory input Obtain mean and variance of estimates Compare mean and variance from step 3 with optimal combinations based on steps 1 and 2 Mean network estimate 93 92 91 90 89 88 87 86 Network performs nearoptimally Mean estimates Estimate variance 9 8 y 9 5 a 7 r w 0 c g s if 39 Iquot 3 E 5 5quot gt g 1 x 4 7 3 4 39n 39 o 3 gquot 3 it 39 a 2 Z 1 O I I I I I I O I I I I I I I I 8687888990919293 0123456789 Mean optimal estimate Optimal variance Same tuning curves same covariance matrices Same tuning curves different covariance matrices x Different tuning curves different covariance matrices Framework for optimal neural computation Stimuli s Assuming a form of neural variability encoding I Probability distributions psquotr Neura lapeitatiioi ia Probabilistic computations fiultitilifi poputllatlii n Output 39erob La39bi l ity distribution i f E I neural activity space probability space Motor actionjudgment Assuming a form of neural variability Conclusions Poisson variability can be generalized to Poisson like variability For Poisson like variability a linear combination of population activity implements optimal cue combination Mappings between Bayesian operations on probability distributions and neural operations can be found For more complex computations a wide open area 9 machine learning algorithms


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