1. Presented by Rhee, Je-Keun
Ch 8. Synaptic Plasticity 8.7 ~ 8.8 Adaptive Cooperative Systems, Martin Beckerman, 1997.
Presented by Rhee, Je-Keun

2. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Contents
8.7 Cortical Response Properties
8.7.1 Circular Environment
8.7.2 Classical Rearing
8.7.3 Orientation Selectivity and Binocular Interactions
8.7.4 Receptive Field Properties
8.8 Mean-Field Network
8.8.1 Mean-Field Approximation
8.8.2 The Cortical Network
(C) 2009, SNU Biointelligence Lab, 

3. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Circular Environment
If the visual fields subtended by cells in the striate cortex are small, then patterned, or contoured, input will resemble noise-corrupted edges oriented in various directions.
a K input pattern set {d1, d2,…, dK}
The environment is then modeled as a circular matrix of inner products of the vectors d1, d2,…, dK.
Assuming that the probabilities p(d = d1) = … = p(d = dK) = 1/K we now have 2K fixed points with selectivities 0, 1/K, 2/K, …, (K-1)/K, and there are K fixed points of maximum selectivity, (K-1)/K, with respect to d.
If the vectors d1, d2,…, dK are orthogonal, or close to orthogonal, or perhaps even far from orthogonal, the K fixed points of maximum selectivity, m1, m2, …, mK, will be stable, and the system will converge to one of these fixed points regardless of its starting point in phase space.
(C) 2009, SNU Biointelligence Lab, 

4. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Classical Rearing
Classical rearing refers to a series of manipulations to kittens visual environment during the period of time during when (modifiable) synaptic geniculocortico and lateral corticocortico connections rapidly adjust their transmission efficiencies in response to changes in the visual stimulus.
The classical rearing conditions include normal rearing (NR), monocular deprivation (MD), reverse suture (RS), strabismus (ST), or artificial squint, binocular deprivation (BD), and the restoration of normal vision following the period of deprived visual input (RE).
(C) 2009, SNU Biointelligence Lab, 

5. Orientation Selectivity and Binocular Interactions
The total input activity to the cortical cells includes contributions from signals produced by external stimuli, from spontaneous activity, and from non-LGN noise.
Assuming that the spontaneous activity is a time-, afferent-, and eye-independent constant dsp
where the subscript a denotes the observed activity.
Consider the two types of afferent stimulation-random noise and patterned activity.
Let us indicate random nose by nj and patterned activity term arising from external input as
(C) 2009, SNU Biointelligence Lab, 

6. Orientation Selectivity and Binocular Interactions
The response of our cortical neurons to spontaneous and patterned activity
where we have appended a term cn(t) for non-LGN noise to the formula for patterned activity.
(C) 2009, SNU Biointelligence Lab, 

7. Receptive Field Properties
In Linsker’s analysis, the receptive fields evolve from a symmetric center-surround form in the outermost layer to a form possessing a measure of orientation selectivity in the deeper layers.
In the study by Kammen and Yuille, the two-dimensional receptive fields, representing a cell’s spatial response function, develop orientation selective properties as consequence of random fluctuation.
Several studies have emphasized that the importance of natural patterned input in developing receptive field properties.
Law and Cooper studies receptive field properties of BCM neurons. In the study receptive field formation was studied within the framework of BCM theory by including retinal processing of natural images.
The sliding modification threshold was defined as
the scale constant τ controls the overall rate of movement of the threshold.
(C) 2009, SNU Biointelligence Lab, 

8. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Mean-Field Network
The BCM theory presented in Section 8.6 was formulated for a single cortical target cell.
In this section we present the extension of the BCM formulation to a network of excitatory and inhibitory neurons.
In the network model, LGN neurons project to cortical excitatory and inhibitory cells, which in turn interact with one another through cortical-cortical synapses.
The cortical-cortical interactions in the approach are mediated by a mean-field in a manner similar to ferromagnetic spins interacting through a Weiss molecular field.
(C) 2009, SNU Biointelligence Lab, 

9. Mean-Field Approximation
The input activity from the LGN is represented by the vector d.
Consider the simple situation where the input from the two eyes are constant in time, and the same for all cortical cells so that di= d for all i. That is, we assume that each cell sees the same portion of the visual field.
The LGN-cortical synaptic weights m become arrays of weights with each row representing a weight vector mi, for the ith cortical cell.
The cortico-cortico synaptic weights are arranged as a matrix L with matrix elements Lij, denoting the connection strengths between axons of cell j and their dendritic targets on cell i.
The activity, or firing rate, of cell i is the sum of contributions from the geniculocortico and cortico-cortico pathways:
(C) 2009, SNU Biointelligence Lab, 

10. Mean-Field Approximation
Introduce the mean activity <c> as the spatially averaged firing rate of all cortical neurons:
The mean-field approximation consists in replacing cj in the cortico-cortico sum by the spatially averaged firing rate to yield
To be consistent, the mean firing rate must satisfy the relation
The firing rate for the ith cortical cells then becomes
(C) 2009, SNU Biointelligence Lab, 

11. Mean-Field Approximation
The difference, i-j, is a measure of neighborness, and is not a function of the absolute positions of the cortical neurons.
If it is assumed that the cortico-cortico synaptic strengths are a function of i-j alone, then Lij becomes a circulant matrix.
In the mean-field approximation, the effect of the cortical network on the output activity is to shift the synaptic weight vector m by the mean-field a:
In this model the network is assumed to be mildly inhibitory: L0<0 with |L0|<1 so that <c> remains positive. We have therefore a model involving excitatory geniculocorical synapses and weakly inhibitory cortico-cortico synapses.
(C) 2009, SNU Biointelligence Lab, 

12. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
The Cortical Network
In the mean-field network, all synapses modify at the same rate.
The model can be generalized to include synapses that modify rapidly and others that modify slowly or not at all.
In this approach we have modifiable (m) and nonmodifiable (z) synapses.
The synaptic evolution equations
(C) 2009, SNU Biointelligence Lab,