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Network models • Firing rate model for neuron as a simplification for network analysis • Neural coordinate transformation as an example of feed-forward neural network • Symmetric recurrent neural networks • Selective amplification, winner-take-all behaviour • Input integration • Receptive field properties of V1 simple cells • Gain modulation to encode multiple parameters (gaze and retinal location) • Sustained activity for short term memory • Associative memory • Excitatory – inhibitory network • Stability analysis and bifurcation • Olfactory bulb
Network models (2k-k^2/n)/2]\approx k links per neuron (n,k) -> (n/2,k)
Neural coordinate transformation Reaching for viewed objects requires transformation from retinal coordinates to body-centered coordinates. A,B: With identical target relative to the body, the image on the retina changes due to gaze change. C: g is gaze angle of eyes relative to head, s is image of object On retina.
Neural coordinate transformation • Visual neurons have receptive fields ‘tied’to the retina. • Left: Motor neurons respond to visual stimuli independent of gaze direction. Stimulus is approaching object from different directions s+g. Three different gaze directions (monkey premotor cortex)
Neural coordinate transformation • Middle: When head is turned but fixation is kept the same (g=-15 degree), the motor neuron tuning curve shifts + 15 degree. The representation is relative to the head.
Neural coordinate transformation • Possible basis for model provided by neurons in area 7a (posterior parietal cortex), whose retinal receptive fields are gain modulated by gaze direction. Left: average firing rate tuning curves for same retinal stimulus at different gaze directions. Right: mathematical model is product of Gaussian in s-x (x=-20o) and sigmoid in g-g (g=20o).
Neural coordinate transformation • Right: results from the model with w(x,g)=w(x+g) with gaze 0o, 10o and –20o (solid, heavy dashed, light dashed) and stimulus at 0o. The shift of the peak in s is equivalent to invariance wrt g+s. • Gain modulated neurons provide general mechanism for combining input signals
Neural integration • Networks in the brain stem of vertebrates responsible for maintaining eye position appear to act as integrators. Eye position changes in response to bursts of ocular motor neurons in brain stem. Neurons in the brainstem integrate these signals. Their activity is approximately proportional to horizontal eye position. • It is not well understood how the brain solves the ‘fine tuning problem of having one of the eigen values exactly 1.
Continuous linear network • A: h(q)=cos(q)+noise and C: its Fourier components hm • B: the network activity v(q) for l=0.9 • D: Fourier components vm. v§ 1=10 h§ 1 and vm=hm otherwise
Orientation tuning in simple cells • Recall that orientation selective cells in V1 could be explained by receiving input from proper constellation of center surround LGN cells. • However, this ignores lateral connectivity in V1, which is more prominent than feed-forward connectivity. • Same as prev. model with h(q)=A(1-e+e cos(2q)) and global lateral inhibition. • Lateral connectivity yields sharpened orientation selectivity. Varying A (illumination contrast) scales the activity without broadening, as is observed experimentally.
Winner take all • When two stimuli are presented to a non-linear recurrent network, the strongest input will determine the response (network details are as previous).
Gain modulation • Adding a constant to the input yields a gain modulation of the recurrent activity. This mechanism may explain the encoding of both stimulus in retinal coordinates (s) and gaze (g) encountered before in parietal cortical neurons.
Sustained activity • After a stimulus (A) has yielded a stationary response in the recurrent network (B), the activity may be sustained (D) by a constant input only (C.).
Associative memory • Sustained activity in a recurrent network is called working or short-term memory. • Long-term memory is thought to reside in synapses that are adapted to incorporate a number ofsustained activity patterns as fixed points. • When the network is activated with an approximationof one of the stored patterns, the network recalls the patterns as its fixed point. • Basin of attraction • Spurious memories • Capacity proportional to N • Associative memory is like completing a familiar telephone number from a few digits. It is very different from computer memory. • Area CA3 of hippocampus andpart of prefrontal cortex
Associative memory • 4 pattern stored in network of N=50 neurons. Two patterns are random and two as shown. • A) Typical neural activity. • B, C) Depending on the initial state one of the patterns is recalled as a fixed point. • Memory degrades with # patterns. • Better learning rules exist • capacity ~ N/(a log 1/a)
Excitatory-Inhibitory networks • MEE=1.25, MIE=1, MII=0, MEI=-1, gE=-10 Hz, gI=10 Hz, tE=10 ms and variable tI. • A) phase plane with nullclines, fixed point and directions of gradients.
Excitatory-Inhibitory networks • B) real and imaginary part of eigenvalue of the stability matrix versus tI. The fixed point is stable up to tI=40 ms and unstable for tI>40 ms.
Excitatory-Inhibitory networks • Network oscillations damp to stable fixed point for tI=30 ms.
Excitatory-Inhibitory networks • For tI=50 ms the oscillations grow. The fixed point is unstable. The dynamics settles in a stable limit cycle, due to the rectification at vE=0. • Such transitions, where the largest real eigenvalue changes sign induce oscilations at finite frequency (6 Hz in this case) is called a Hopf bifurcation.
Olfactory bulb • Olfaction (smell) is accompanied by oscillatory network activity. • A) During sniffs the activity of the network increases and starts to oscillate. • B) Network model 7.12-13 with MEE=MII=0. hE is the external input that varies with time. hI is positive top-down input from cortex.
Olfactory bulb • A) Activation functions F assumed in the model. • B) h_E changes the stability of the stable fixed point at low network activity. Largest real eigenvalue crosses 1 around t=100 ms inducing 40 Hz oscillations. Oscillations stop around 300 ms.
Olfactory bulb The role of h_E is twofold: • it destabilizes the fixed point of the whole network inducing network oscillations • Its particular input to different neurons yields different patterns for different odors