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Oscillations in the Olfactory Bulb: Li and Hopfield’s Model Ranit Fink Anthony Sanfiz Simon Fischer-Baum

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Outline of the Olfactory System

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Physiology of the Olfactory Bulb What is in the bulb? –Glomeruli –Mitrial –Granule –Tufted –SA –PG Dendro-dendritic connections From Li and Hopfield (1989)

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Circuitry of the Olfactory Bulb Inputs from the Olfactory Receptors (I) Inputs from the Cortex (I c )Bulbar Outputs Glomeruli Layer Mitrial Layer Granule Layer

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Oscillations Intrinsic to bulb itself –Continues even in central inputs are cut off Input odor not necessary EEG shows that oscillation arises during inhalation and stops early in exhalation Specific odors have specific oscillation patterns From Freeman and Schneider (1982)

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General Model Assumptions Mitrial / Glomeruli considered one unit –Glomeruli Uniformity –Glomeruli Selectivity Granule:Mitrial = 1:1, not 200:1 –More complicated P odors –Increase strength of granule cell synapses 25,000 Receptors 1 Glomerulus 25 Mitrial Cells 200 Granule per 1 Mitrial Convergence and Divergence in the Olfactory Bulb

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General Model Assumptions Tufted cells considered Mitrial cells – Mitrial and Tufted cells have no functional distinction Inhibitory Neurons Neglected (or not) –Intraglomeruli inhibition helps maintain uniformity –Interglomeruli inhibition heightens contrast

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Model Receptor Firing Patterns From Getchell and Shepherd (1982) Li and Hopfield (1989)

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Model Input To the Mitrial cells I = I odor + I background | I odor > 10 or 20 * I background P odor,i (t – t 0 ) + I odor,i (t 0 ) t 0 < t < t 1 I odor,i (t) = I odor,i (t 1 ) * e (t- t0)/tau t > t 0 t0 = Starting time of inhale, t1 = Starting time of exhale, tau = 33msec const, how we fast we go down I background = Background noise, a constant or Gaussian noise (Held or centered at 0.243). P odor = Pattern of selectivity for each mitrial cell to specific input odor.

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Model Input To the Granule cells I c = A constant with the value 0.1 or Gaussian noise center at 0.1 Constants chosen to be close to threshold Reflect weak oscillatory activity with no input (Freeman and Schneider 1982). Sniff cycle lasts ms to model rabbit olfaction

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Input to Output States From Freeman and Skarda 1985 From Li and Hopfield 1989

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Inside the Cell We define Mitral cell as X = {x1, x2, …xN} We define Granule cell as Y = {Y1, Y2, …YN} S x ’ + S x ’ * tanh((x - th)/ S x ’ ) x < th g x (x) = S x ’ + S x * tanh((x - th)/ S x ) x >= th S y ’ + S y ’ * tanh((y - th)/ S y ’ ) y < th g y (y) = S y ’ + S y * tanh((y - th)/ S y ) y >= th th = threshold with value 1. S x ’, S x, S y ’, S y scalers 0.14,1.4,0.29,2.9 respectively

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Ho=Ho= Wo=Wo=

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“Box” For the Olfactory Bulb X’(t + ∆t) =– H 0 * G y (Y(t)) – α x X(t) + I –-H 0 * G y (Y) = Influence of Granule cells on Mitral cells (negative due to inhibition) –α x X= friction. α x = 1/tau x where tau x = 7 –I = the total input, including the noise Y’(t + ∆t) = W 0 * G x (X(t)) – α y Y(t) + I c –W 0 * G y (Y) = Influence of Mitrial cells on Granule cells –α y X= friction. α y = 1/tau y where tau y = 7 –I c = the total input, including the noise i

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Interactions Between Cells The communication between the cells are defined by two matrices. We define them by H 0 and W 0. –H 0 describes the synaptic connections from mitral to granule cells. –W 0 describes the synaptic connections from granule to mitral cells. The i th cell of mitral compatible to the granule (i * (m/n)) th cell –Map the 2D shape of the cells in real life to a 1D ring, inside are mitral cells and outside are granule cells.

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