Neural Network Models in Vision Peter Andras peter.andras@ncl.ac.uk

The Visual System R LGN V1 V3 V2 Lower V5 V4 Higher

Neurons Rod Horizontal Bipolar Amacrine Ganglion

Neuron Models The McCullogh-Pitts model x1 x2 x3 … xn-1 xn w1 Output Inputs y w3 . . . wn-1 wn

Neuron Models K+ Na+ Na+ Na+ The Hodgkin-Huxley Model Na+ K+ K+ K+ Na+

Modelling Methodology Physiological measurements Electrode Response Stimulus Other methods: EEG, MRI, PET, MEG, optical recording, metabolic recording

Modelling Methodology Response characterisation in terms of stimulus properties Stimulus

Modelling Methodology Models: A. Statistical models: large number of neurons, with a few well-defined properties, the response is analysed at the population level;

Modelling Methodology Models: B. Macro-neural models: simplified model neurons organised in relatively simple networks, the overall input-output relationship of the full network is analysed;

Modelling Methodology C. Micro-neural models: the neurons are modelled with many details and models of individual neurons or networks of few detailed neurons are analysed. Models:

Modelling Methodology Physiological measurements Response characterisation Model selection OBJECTIVE 1: match the measured response properties by the response properties of the model. OBJECTIVE 2: test the theories, generate predictions.

Neural Network Models Retina: ON and OFF centre ganglion cells Bipolar cells +1 -1 ON OFF Preferred stimulus

Neural Network Models Retina: ON and OFF centre ganglion cells Measured response of an ON cell The response of a model ON cell

Neural Network Models V1: Orientation selective cells LGN cells Preferred stimulus

Neural Network Models V1: Orientation selective cells Measurement

Neural Network Models V1: Ocular dominance patterns and orientation maps

Neural Network Models V1: Ocular dominance patterns and orientation maps Neuron = Feature vector: orientation preference; spatial frequency; eye preference; temporal frequency; Training principles: the neuron fires maximally when the stimulus matches its preferences set by the feature vector; the neuron fires if its neighbours fire; when the neuron fires it adapts its feature vector to the received stimulus.

Neural Network Models V1: Ocular dominance patterns and orientation maps Mathematically: Neurons: (wi , ci); wi – feature vector; ci – position vector; Training set: xt , training vectors, they have the same dimensionality as the feature vectors; Training: i* = index of the neuron for which d(wi*, xt) < d(wi, xt), for every i  i*; wi = (1-) wi +  xt , for all neurons with index i, for which d(ci, ci*) < .

Neural Network Models V1: Contour detection Stimulus

Neural Network Models V1: Contour detection Neural interactions: specified by interconnection weights. Mechanism: constraint satisfaction by mutual modification of the firing rates. Result: the neurons corresponding to the contour position remain active and the rest of the neurons become silent.

Neural Network Models V5: Motion direction selective cells Orientation selective cells delay effect -1 +1 Preferred stimulus

Neural Network Models Visual object detection Object Invariant combination of features Features: colour; texture; edge distribution; contrast distribution; etc. Object detection

Neural Network Models Visual object detection Method 1: Hierarchical binary binding of features Colour Texture Edges This method leads to combinatorial explosion. Contrast

Neural Network Models Visual object detection Method 2: Non-linear segmentation of the feature space. Colour Texture Edges Learning by back-propagation of the error signal and modification of connection weights. Contrast

Neural Network Models Visual object detection Method 3: Feature binding by synchronization.

Critical Evaluation Neural network models typically explain certain selected behavioural features of the modelled neural system, and they ignore most of the other aspects of neural activity. These models can be used to test theoretical assumptions about the functional organization of the neurons and of the nervous system. They provide predictions with which we can determine the extent of the validity of the model assumptions. One common error related to such models is to invert the causal relationship between the assumptions and consequences: i.e., the fact that a model produces the same behavior as the modelled, does not necessarily mean that the modelled has exactly the same structure as the model.

Revised View of the Neural Network Models Revised interpretation: neurons = anatomical / functional modules (e.g., cortical columns or cortical areas); connections = causal relationships (e.g., activation of bits of LGN causes activation of bits of V1); activity function of a neuron = conditional distribution of module responses, conditioned by the incoming stimuli;

Revised View of the Neural Network Models Bayesian network model x1 f1(x1) P(y1|x1) y1 y1 x1 x2 f2(x2) y2 P(x1, x2, x3, x4) f(y1, y2, y3, y4) P(y2|x2) y2 P(y | y1, y2, y3, y4) x2 y y y3 x3 y3 x3 f3(x3) P(y3|x3) yi = fi(xi) y = f(y1, y2, y3, y4) P(x1, x2, x3, x4) P(yi | xi) P(y | y1, y2, y3, y4) y4 x4 y4 x4 f4(x4) P(y4|x4)

Revised View of the Neural Network Models Advantages of the Bayesian interpretation: relaxes structural restrictions; makes the models conceptually open-ended; allows easy upgrade of the model; allows relaxed analytical search for minimal complexity models on the basis of data; allows statistically sound testing;

Conclusions Neuron and neural network models can capture important aspects of the functioning of the nervous system. They allow us to test the extent of validity of the assumptions on which the models are based. A common mistake related to neural network models is to invert the causal relationship between assumptions and consequences. This can lead to far reaching conclusions about the organization of the nervous system on the basis of natural-like functioning of the neural network models that are invalid. The Bayesian reinterpretation of neural network models relaxes many constraints of such models, makes their upgrade and evaluation easier , and prevents to some extent incorrect interpretations.

Seminar Papers 1. PNAS, 93, 623-627, Jan. 1996 2. PNAS, 96, 10530-10535, Aug. 1999