Neural Computation Theory Seminar: Exploring Hopfield Networks

1. Artificial Neurons: Hopfield Networks Introduction Neurophysiological Background Modeling Simplified Neurophysiological Information The Hopfield Model The Associative Memory Problem The Model Updating rules One Pattern Many Patterns Stability of a particular pattern Storage Capacity The Energy Function Discussion on Philosophy and Methodology Seminar: Introduction to the Theory of Neural Computation

2. Introduction Inspiration on today’s research in neural computation comes from neuroscience and is largely motivated by the possibility of modeling artificial computing networks. So models are extremely simplified when seen from a neurophysiological point of view, but one should gain insight in the behaviour of “biological” networks. First the neurophysiological background should be described: information for modeling simplified neurophysiological processes description and the behaviour of neural networks – Hopfield Networks. Seminar: Introduction to the Theory of Neural Computation

3. Neurophysiological Background • basic elements for a neural network: • neurons and their connections • Systematically the nervous system can be divided into three parts • Input • central processing unit • output • In the field of ANNs, networks will be constructed from neurons which have the canonical division into an input part • (dendritic arbor), a processing part (soma) and a signal transmission part (axon). Seminar: Introduction to the Theory of Neural Computation

4. Modeling Simplified Neurophysiological Information (1) Seminar: Introduction to the Theory of Neural Computation

5. Modeling Simplified Neurophysiological Information (2) Seminar: Introduction to the Theory of Neural Computation

6. Modeling Simplified Neurophysiological Information (3) This operation and its components leads to the basic formular The operation can be expressed by the logical truth function is a function which is 1 if the statement in the square brackets is true and is 0 otherwise indicades, whether a spike (1 is sent) will appear in the output axon defines variables which are themselves zeros and ones (which can also be considered as truth functions of some statement ) Seminar: Introduction to the Theory of Neural Computation

7. Modeling Simplified Neurophysiological Information (3) A significant leap is acomplished, when the multi-neuron (multi-perceptron) is closed onto itself, where the neurons form a feedback mechanism. ! An ANN is no longer a linear, but a dynamical system, when output axons (signal transmission parts) become input channels, there is a time shift. Seminar: Introduction to the Theory of Neural Computation

8. Overview Introduction Neurophysiological Background Modeling Simplified Neurophysiological Information The Hopfield Model The Associative Memory Problem The Model Updating rules One Pattern Many Patterns Stability of a particular pattern Storage Capacity The Energy Function Discussion on Philosophy and Methodology Seminar: Introduction to the Theory of Neural Computation

9. The Hopfield Model - The Associative Memory Problem Hopfield networks consist of the previously described elements and are totally dynamical, so including the time shift and possible updating rules. ! basic problem: to store a set of p patterns in such a way that when presented with a new pattern , the network responds by producing whichever one of the stored patterns most closely resembles The space of all possible states of the network, is called the configuration space. basins of attraction: Division of the the confirguration space by stored patterns Seminar: Introduction to the Theory of Neural Computation

10. The Model The dynamics of the network can be represented by: whereis represented forwith the conversion from =0 or 1 via =2 -1 and sgn(x) is defined by: The threshold terms can be dropped in consideration on random patterns being used. Seminar: Introduction to the Theory of Neural Computation

11. Updating rules - Two simplified versions Synchronous or Parallel All neurons update their activity states simultaneously at discrete time steps n, where n = 1, 2, …, as if governed by a clock. The inputs of every neuron in the network are determined by the same activity state of the network in the time interval (n-1) < t < n. This choice requires a central clock or pacemaker and is sensitive to timing errors. Asynchronous or Sequential (more natural for both brains and artificial networks) All neurons are updated one by one, where one can proceed in either of two ways: at each time step, select at random a unit i to be updated and apply the rule let each unit independently choose to update itself, with some constant probability per unit, according to In this mode: every neuron coming up for a decision has full information about all the decisions of the individual neurons that have been updated before it. Seminar: Introduction to the Theory of Neural Computation

12. One Pattern The condition for one pattern which should be memorized is For constant of proportionality, using 1/N: = 1/N If fewer then half of the bits of the starting patterns are wrong they will be overwhelmed in the sum for the net input The network will correct errors and so the pattern is an attractor All starting configurations with more than half the bits different from the original pattern will end up in the reversed state - , which leaves to a symmetrically divided configuration spaces into two basins of attraction. Seminar: Introduction to the Theory of Neural Computation

13. Many Patterns ! hypothesis made by Hebb (1949): changes proportional to the correlation between the firing of the pre- and post-synaptic neurons achieved through: applying the set of patterns to the network during the training phase adjust the strenghts according to such pre/post correlations Seminar: Introduction to the Theory of Neural Computation

14. Overview Introduction Neurophysiological Background Modeling Simplified Neurophysiological Information The Hopfield Model The Associative Memory Problem The Model Updating rules One Pattern Many Patterns Stability of a particular pattern Storage Capacity The Energy Function Discussion on Philosophy and Methodology Seminar: Introduction to the Theory of Neural Computation

15. Stability of a particular pattern (1) Going back to the condition for a stable one pattern and the definiton of the net input the stability condition generalizes to Taking the net input to unit in pattern v is seperating the sum on into the special term = v ! Seminar: Introduction to the Theory of Neural Computation

16. Stability of a particular pattern (2) Meaning crosstalk term (is less than 1, in most cases) ! If the second term were zero, one can conclude that pattern number v was stable according to This is still true if the second term is small enough: if its magnitude is smaller than 1 it cannot change the sign of Seminar: Introduction to the Theory of Neural Computation

17. Storage Capacity One consider the quantity by The just depend on the patterns that one attempt to store The distribution of values for the crosstalkterm ! For p random patterns and N units this is a Gaussian with variance The shaded area is , the probability of error per bit Seminar: Introduction to the Theory of Neural Computation

18. The Energy Function (1) … was atopted from a physical analogy to magnetic systems into the neural network theory and is one of the most important contributions of the Hopfield paper. One can imagine an energy landscape metaphor “above” the configuration space with a multi-dimensional surface with hills and valleys. The energy function is Seminar: Introduction to the Theory of Neural Computation

19. The Energy Function (2) central property: It is a function that always decreases (or remains constant) as the system evolves according to its dynamical rule. The attractors are at local minima (the valleys) of the energy surface, the dynamics then can be thought of as similar to the motion of a partical on the energy surface under the influence of gravity (pulling it down) and friction (so that it does not overshoot). Seminar: Introduction to the Theory of Neural Computation

20. using: ! analog: many-pattern case: patterns should be made into local minima of H The Energy Function (3) alternate derivation of the Hebb prescription (as we know it from the many pattern case) minimized when the overlap between network configuration and the stored pattern (one pattern case) is largest. Seminar: Introduction to the Theory of Neural Computation

21. The Energy Function (4) Multplying out out leads to the original energy function ! good approach of finding the appropriate connection strength , by finding an energy function whose minimum satisfies a problem of interest, and by multiplying it out Seminar: Introduction to the Theory of Neural Computation

22. The Energy Function (5) “Simple and nice” proving of the central property of the Energy Function It is a function that always decreases (or remains constant) as the system evolves according to its dynamical rule. ! energy function for the t state energy function for the t+1 state Seminar: Introduction to the Theory of Neural Computation

23. Discussion on Philosophy and Methodology (1) • Research in these particular areas involves many different fields of science: • Biology • chemistry • physics • (...) • natural phenomena are described by mathematical models, sometimes being interpreted that all natural phenomena are reducible to physical laws. • Alternatively - as I would say too - reduction can be given a very intuitive sense in which it not only exists but is extremely useful and productive. • Hopflied once stated that “the brain is a physical system”, which may indeed sound like a call for a reduction of thought process, nevertheless concepts originating in physics can be used as analogues, including energy, field, relaxation etc. Seminar: Introduction to the Theory of Neural Computation

24. Discussion on Philosophy and Methodology (2) The theory of attractor neural networks (ANN) has engaged in providing a minimal amount of propositions which can be confronted with experiment. This matter plays a role in discussing the attitude to verification and/or falsification and the fact that a theoratical framework must be fended off by an explanation. In many instances systems have been constructed (hardware implemantations/ computer simulations), being experimental setups for described models and providing a truly impressive agreement on predicitions by the analysis of the models But this will not please no experimenter who records, using ingeniuos techniques, the electrical activities in the cortex of cats or monkeys, for example. For the future: the theory of neural networks is to produce models, about cognitive processes and which should be robust to the type of disorder, fluctuations, disruptions one can imagine the brain to be operating under. Including: parallel processing or potential for abstraction ! Seminar: Introduction to the Theory of Neural Computation

25. Discussion on Philosophy and Methodology (3) So what happenes if a experiment may not show the type of bahaviour identified as the emergent dynamics. Interpretion: a refutation of the theoretical construction or arguing that the experiment has missed the theory. ! Thank you very much! Feel free to ask questions! Seminar: Introduction to the Theory of Neural Computation