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N. Laskaris Professor John Hopfield The Howard A. Prior Professor of Molecular Biology Dept. of Molecular Biology Computational Neurobiology; Biophysics Princeton University The physicist Hopfield showed that models of physical systems could be used to solve computational problems Such systems could be implemented in hardware by combining standard components such as capacitors and resistors. The importance of the Hopfield nets in practical application is limited due to theoretical limitations of the structure, but, in some cases, they may form interesting models. Usually employed in binary-logic tasks : e.g. pattern completion and association The concept In the beginning of 80s Hopfield published two scientific papers, which attracted much interest. (1982): ‘’Neural networks and physical systems with emergent collective computational abilities’’. Proceedings of the National Academy of Sciences, pp. 2554-2558. (1984): ‘’Neurons with graded response have collective computational properties like those of two-state neurons’’. Proceedings of the National Academy of Sciences, pp. 81:3088-3092 This was the starting point of the new era of neural networks, which continues today ‘‘The dynamics of brain computation” The core question : How is one to understand the incredible effectiveness of a brain in tasks such as recognizing a particular face in a complex scene? Like all computers, a brain is a dynamical system that carries out its computations by the change of its 'state' with time. Simple models of the dynamics Using these collective properties of neural circuits are described in processing information that have collective dynamical properties. is effective in that it exploits the spontaneous properties These can be of exploited nerve cells and circuits in recognizing sensory patterns. to produce robust computation. J. Hopfield’s quest While the brain is totally unlike modern computers, Associative memory,as computation. much of what it does can be described logic and inference, His research focuses recognizing an odor or a chess position, on understanding parsing the world into objects, andhow generating appropriate sequences of brain locomotor the neural circuits of the muscle commands produce are all describable such powerful and complex as computation. computations. Olfaction However,problem olfaction remote sensing, The simplest inallows olfaction and much morehas complex computations is simply identifying a known odor. Hopfield been studying windmight direction how such involving computations be performed and mixtures of odors by fluctuating the known neural circuitry must be described to account for the ability of the olfactory bulb ofand homing pigeonscortex or slugs navigate prepiriform of to mammals throughcircuits the useof ofsimpler odors. animals. or the analogous Dynamical systems Any computer does its computation Systems of differential equations by its changes in internal state. can represent these aspects of neurobiology. In neurobiology, He seeks tothe understand aspects change ofsome potentials ofof neurons neurobiological computation (and changes in the strengths of the synapses) through the behavior ofthe equations withstudying time is what performs computations. modeling the time-evolution of neural activity. Action potential computation For much of neurobiology, information is represented by the paradigm of ‘‘firing rates’’, i.e. information is represented by the rate of generation of action potential spikes, and the exact timing of these spikes is unimportant. Action potential computation Since action potentials last only about a millisecond, the use of action potential timing seems a powerful potential means of neural computation. Action potential computation There are cases, for example the binaural auditory determination of the location of a sound source, where information is encoded in the timing of action potentials. Speech Identifying words in natural speech is a difficult computational task which brains can easily do. They use this task as a test-bed for thinking about the computational abilities of neural networks and neuromorphic ideas Simple (e.g. binary-logic ) neurons are coupled in a system with recurrent signal flow A 2-neurons Hopfield network of continuous states characterized by 2 stable states 1st Example Contour-plot 2nd Example A 3-neurons Hopfield network of 23=8 states characterized by 2 stable states The behavior of such a dynamical system is fully determined by the synaptic weights 3rd Example Wij = Wji And can be thought of as an Energy minimization process Hopfield Nets are fully connected, symmetrically-weighted networks that extended the ideas of linear associative memories by adding cyclic connections . Note: no self-feedback ! Operation of the network After the ‘teaching-stage’, in which the weights are defined, the initial state of the network is set (input pattern) and a simple recurrent rule is iterated Regarding training a Hopfield net There are two of operation: till convergence to a modes stable state (output pattern) as a main content-addressable memory Synchronous vs. Asynchronous updating the outer-product rule for storing patterns is used Hebbian Learning Probe pattern Dynamical evolution A Simple Example Step_1. Design a network with memorized patterns (vectors) [ 1, -1, 1 ] & [ -1, 1, -1 ] Step_2. Initialization There are 8 different states that can be reached by the net and therefore can be used as its initial state #1: y1 #2: y2 #3: y3 Step_3. Iterate till convergence Synchronous Updating 3 different examples of the net’s flow It converges immediately Step_3. Iterate till convergence - Synchronous Updating - Stored pattern Schematic diagram of all the dynamical trajectories that correspond to the designed net. Or Step_3. Iterate till convergence - Asynchronous Updating Each time, select one neuron at random and update its state with the previous rule and the –usual- convention that if the total input to that neuron is 0 its state remains unchanged Explanation of the convergence There is an energy function related with each state of the Hopfield network E( [y1, y2, …, yn]T ) = -Σ Σ wij yi yj where [y1, y2, …, yn]T is the vector of neurons’ output, wij is the weight from neuron j to neuron i, and the double sum is over i and j. The corresponding dynamical system evolves toward states of lower Energy States of lowest energy correspond to attractors of Hopfield-net dynamics E( [y1, y2, …, yn]T ) = = -Σ Σ wij yi yj Attractor-state Capacity of the Hopfield memory In short, while training the net (via the outer-product rule) we’re storing patterns by posing different attractors in the state-space of the system. While operating, the net searches the closest attractor. When this is found, the corresponding pattern of activation is outputted How many patterns we can store in a Hopfield-net ? 0.15 N, N: # neurons Computer Experimentation Class-project A simple Pattern Recognition Example Stored Patterns (binary images) Perfect RecallImage Restoration Erroneous Recall Irrelevant results Note: explain the ‘negatives’ …. The continuous Hopfield-Net as optimization machinery [ Tank and Hopfield ; IEEE Trans. Circuits Syst. 1986; 33: 533-541.]: ‘Simple "Neural" Optimization Networks: An A/D Converter, Signal Decision Circuit, and a Linear Programming Circuit’ Hopfield modified his network so as to work with continuous activation and -by adopting a dynamical-systems approach- showed that the resulting system is characterized by a Lyaponov-function who termed it ‘Computational-Energy’ & which can be used to tailor the net for specific optimizations The system of coupled differential equation describing the operation of continuous Hopfield net dui ui n Tij g j ( u j ) I i dt ni j 1 Neuronal outputs: Yi ≡ Vi Biases: Ii Weights: Wij ≡ Tij g( u) 1 n n E Tij g i ( u i ) g j ( u j ) 2 i 1 j1 1 n n E Tij Vi Vj 2 i 1 j1 n 1 2 1 tanh(gain u) n I i gi (ui ) i 1 Tij=Tji και Tij=0 I i Vi The Computational Energy i 1 When Hopfield nets are used for function optimization, the objective function F to be minimized is written as energy function in the form of computational energy E . The comparison between E and F leads to the design, i.e. definition of links and biases, of the network that can solve the problem. The actual advantage of doing this is that the Hopfield-net has a direct hardware implementation that enables even a VLSI-integration of the algorithm performing the optimization task An example: ‘Dominant-Mode Clustering’ Given a set of N vectors {Xi} define the k among them that form the most compact cluster {Zi} N N 2 F ({ ui } ) = ui u j Xi - X j i=1 j=1 1 if Xi { Zi } {u i } with ui = k : {u i } i 0 if Xi { Zi } N The objective function F can be written easily in the form of computational energy E There’s an additional Constraint so as k neurons are ‘on’ N F ({ ui } ) = N Xi -X j i=1 2 ui u j j=1 N 1 N N F= - T ij V i V j - I i V i 2 i=1 j=1 i=1 0 if i j 2 T ij = T ji = -2 D(i, j)= 2 X i X j I iobj = 0 With each pattern Xi we associate a neuron in the Hopfield network ( i.e. #neurons = N ). The synaptic weights are the pairwise-distances (*2) If its activation is ‘1’ when the net will converge the corresponding pattern will be included in the cluster. A classical example: ‘The Travelling Salesman Problem’ The principle Coding a possible route as a combination of neurons’ firings 53 4 1 2 5 |5-3|+|3-4|+|4-1|+|1-2|+|2-5| An example from clinical Encephalography The problem : The idea : The solution : ‘‘Hopfield Neural Nets for monitoring Evoked Potential Signals’’ N. Laskaris et al. [ Electroenc. Clin. Neuroph. 1997;104(2) ] The Boltzmann Machine Improving Hopfield nets by simulating annealing and adopting more complex topologies (430 – 355) π.X. ‘Ας κλείσω λοιπόν εδώ . . . . .............. . . . . κάποιος άλλος, ίσως θα συμπληρώσει όσα δεν μπόρεσα να ολοκληρώσω’ - Θεμιστογένης ο Συρακούσιος 1ο έτος της 105ης Ολυμπιάδας ΕΛΛΗΝΙΚΑ (1979-1982) (1982) Hopfield-nets PNAS ‘‘ Τα παιδιά στην Κερκίδα είναι η μόνη σου Ελπίδα ....’’ A Very Last Comment on Brain-Mind-IntelligenceLife-Happiness How I Became Stupid by Martin Page Penguin Books, 2004, 160 pp. ISBN: 0-14-200495-2 In HOW I BECAME STUPID, The 25-year-old Antoine concludes ‘‘to think is to suffer’’, a twist on the familiar assertion of For Antoine, intelligence Descartes. is the source of unhappiness. He embarks on a series of hilarious strategies to make himself stupid and possibly happy Animals that Abandon their Brains Dr. Jun Aruga Laboratory for Comparative Neurogenesis A “primitive but successful” animal Oxycomanthus japonicus There is astonishing diversity in the nervous systems of animals, and the variation between species is remarkable. From the basic, distributed nervous systems of jellyfish and sea anemones to the centralized neural networks of squid and octopuses to the complex brain structures at the terminal end of the neural tube in vertebrates, the variation across species is humbling people may claim that “more advanced” species like humans are the result of an increasingly centralized nervous system that was produced through evolution. This claim of advancement through evolution is a common, but misleading, one. It suggests that evolution always moves in one direction: the advancement of species by increasing complexity evolution may selectively enable body structures that are more enhanced and complicated, but it may just as easily enable species that have abandon complex adaptations in favour of simplification. Brains, too, have evolved in the same way. While the brains of some species, including humans, developed to allow them to thrive, others have abandoned their brains because they are no longer necessary. For example, the ascidian, or sea squirt, lives in shallow coastal waters and which is a staple food in certain regions, has a vertebrate-like neural structure with a neural tube and notochord in its larval stage. As the larvae becomes an adult, however, these features disappear until only very basic ganglions remain. In evolutionary terms this animal is a “winner” because it develops a very simplified neural system better adapted to a stationary life in seawater In the long run, however, evolutionary success will be determined by what species survives longer: humans with their complex brains (and their weapons) or the brainless Dicyemida 1948-1990 Δισέγγονος του Ζορμπά και ανηψιός της Ελλης Αλεξίου. Γεννήθηκε στην Αθήνα. Ξεκίνησε την καριέρα του το 1970 από τη Θεσσαλονίκη με το συγκρότημα-ντουέτο "Δάμων και Φιντίας". Το 1976 ιδρύει το συγκρότημα "Σπυριδούλα". Η σκέψη μας είναι το αφεντικό ή ο υπηρέτης μας ; Emotional Intelligence also called EI or EQ , describes an ability, capacity, or skill to perceive, assess, and manage the emotions of one's self, of others, and of groups H ποιητική νοημοσύνη μπορεί να λείπει από τους παντογνώστες, κι ωστόσο να κατοικεί μέσα στον πιο απλόν άνθρωπο Class-project Oral-Exams Oral-Exam Appointments Date AEM 1st hour Time 2nd hour 3rd hour 31 May 5 June 7 June 794 845 893 899 915 920 932 949 1023 711 809 874 909 923 950 979 1024 1227 627 887 946 960 962 980 995 1202 1223 Further Inquiries