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LECTURE 6 Network Models I. Introduction − Basic concepts of neural networks II. Realistic neural networks − Homogeneous excitatory and inhibitory populations − The olfactory bulb − Persistent neural activity What is a ( an Artificial) Neural Network? • Inspired from real neural systems • Having a network structure, consisting of artificial neurons (nodes) and neuronal connections (weights ) • A general methodology for function approximation • It may either be used to gain an understanding of biological neural networks, or for solving artificial intelligence problems How neural systems look like? • Neuron: the fundamental singalling/computational units • Synapses: the connections between neurons • Layer: neurons are organized into layers • Extremely complex: around 1011 neurons in the brain, each with 104 connections Imagine 6 business cards putted together. − Human: as large as a dinner napkin; − Monkey: about the size of a businessletter envelope; − Rat: a stamp. There are about 105 neurons within 1mm2. 2 mm Cortical columns: − In the neocortex, neurons lie in 6 vertical layers highly coupled within cylindrical columns with about 0.1 to 1 mm in diameter (e.g. ocular dominance columns and orientation maps in V1) Three main classes of interconnections (e.g. visual system): − Feedforward connections bring input to a given region from another region located at an earlier stage along a particular processing pathway − Recurrent synapses interconnect neurons within a particular region that are considered to be at the same stage along the processing pathway − Top-down connections carry signals back from areas located at later stages. Compare neural systems with computers The brain is superior than modern computers in many aspects (e.g., face recognition and talk). • A different style of computation: parallel distributed processing • An universal computational architecture: the same structure carries out many different functions • It can learn new knowledge, therefore it is adaptive The idea of artificial neural networks 1 Nodes --- Neurons (artificial neurons, performing a linear or non-linear mapping) 2 Weights --- Synapses 3 Mimic the network structure of neural systems • A single neuron’s function is simple. The specific functions come from the network structure • ANN is more and more engineering-driven nowadays, its biological root is gradually losing. The key of ANN is on the design and training of suitable network structures • The connection style – Feed-forward and recurrent Feedforward and recurrent networks An example of one-layer feed-forward neural network x1 x2 w1 w2 b y Activation function ‘+’ or ‘-’ n w3 y f ( wi xi b) i 1 wn xn Bias ‘b’ is an external parameter that can be modeled by adding an extra input Activation function • Linear-Threshold function: if V Vth 0 g (V ) aV , a 0 if V Vth f V • Sigmoid function: g 1 g (V ) 1 e V Sigmoid function The sigmoid function 1.0 0.0 -6.0 0.0 6.0 1 with β = 1.0 V 1 e -6.0 0.0 β = 5.0 6.0 1.0 0.0 -6.0 0.0 β = 0.1 6.0 -25.0 0.0 β = 0.1 25.0 An example of 3-layer feed-forward network x1 x2 y1 y2 ym n yi f ( wij x j bi ) xn Input Layer j 1 Hidden layers Output Layer Neural Network (NN) for function approximation NN transforms input into output. In other words, a NN model implements a function NN has a set of adjustable parameters (weights, activation threshold, and etc.) Network parameters (in particular, network weights) are free to be adjusted in order to achieve desired functions Type of learning used depends on task at hand • Supervised learning: have a teacher • Unsupervised learning: no teacher • Reinforcement learning: no detailed instruction, only the final reward is available Possible use of artificial neural networks (Do you remember the application of Neurocomputing? ) Pattern recognition (face recognition, radar systems, object recognition), Sequence recognition (gesture, speech, handwritten text recognition) Medical diagnosis, Financial applications, Data mining, Stock market analysis, Weather forecast • Classification: given an input, decide its class index • Regression: given an input, decide the corresponding continuous output value • I will mainly focus on the realistic neural networks in this course • And Prof. Shi Zhongzhi will covers the artificial neural networks and … I. Introduction − Basic concepts of neural networks II. Realistic neural networks − Homogeneous excitatory and inhibitory populations − The olfactory bulb − Persistent neural activity − …… Homogeneous excitatory and inhibitory populations (i.e. Wilson-Cowan model) • It describes the dynamics of interacting excitatory and inhibitory neuronal populations • Usually two neurons (excitatory/inhibitory) • Explain the oscillation source in neural systems • A typical recurrent network wEI wII I Input E E wEE Input I wIE dvE E vE g ( wEE vE wEI vE E ) Compare with dt dvI I vI g ( wII vI wIE vI I ) dt dV m EL V Rm I e dt • Here g(x) is an activation function or the steady-state firing rate as a function of input • g(x) is Linear-Threshold function: g 0 g ( x) x if x 0 if x 0 x dvE [vE g (1.25vE vI 10)] / E dt dvI [vI g (vE 10)] / I dt E 10ms adjustable parameter 1. Find the fixed points by setting: dv E 0 dt dv I 0 dt VE = 26.67Hz VI = 16.67Hz 2. Identify stability by using phase plane or 2-D phase space portrait y (x(t+h),y(t+h)) =(x(t)+hdx/dt, y(t)+hdy/dt) dVI /dt < 0 (x(t),y(t)) x dVI /dt > 0 Adjust parameter I 30ms (Dayan and Abbott, 2001) Adjust parameter I 50ms (Dayan and Abbott, 2001) Hopf bifurcation As a parameter is changed, a fixed point of a dynamical system loses stability and a limit cycle emerges I. Introduction − Basic concepts of neural networks II. Realistic neural networks − Homogeneous excitatory and inhibitory populations − The olfactory bulb − Persistent neural activity − …… The olfactory bulb InputE InputI InputI Input dvE E vE FE ( wEE vE wEI vE Input E ) dt dvI I vI FI ( wII vI wIE vI Input I ) dt where E I 6.7ms, wEE wII 0, and n m 10 • A recurrent network • A ?-D phase space is needed • The InputE, must induce a transition between fixed-point and oscillatory activity Activities of four of ten mitral and granule cells during a single sniff cycle for two different odors (Dayan and Abbott, 2001) (Dayan and Abbott, 2001) I. Introduction − Basic concepts of neural networks II. Realistic neural networks − Homogeneous excitatory and inhibitory populations − The olfactory bulb − Persistent neural activity Persistent neural activity • It refers to a sustained change in action potential discharge that long outlasts a stimulus • It is found in a diverse set of brain regions and organisms and several in vitro systems, suggesting that it can be considered a universal form of circuit dynamics Several examples for persistent neural activity • Oculomotor neural integrator cells in an awake behaving goldfish • Monkey prefrontal cortical cells • Head direction cells in rat An oculomotor neural integrator cell in an awake behaving goldfish (Major and Tank 2004) Generation mechanism of persistent neural activity during memory-guided saccade task in the prefrontal cortex E dV jE EE ij w dt i j w ri e s E I S V jE w EE r r 0 . 5 r ji i j EE max e ( pj pi ) 2 2a2 (T pi ) 2 2b 2 I dV I E ri dt i (300ms) 120 units Persistent activity Activation function: r 1 1V jE 1 e A distractor input for 300ms g ( ) (Gruber et al. 2006) From this PFC network, we learn: 1. The persistent PFC activity is a network effect associated with recurrent excitation 2. The network has a line (or ring) attractor Homework 1. 什么是动力系统的 Hopf 分叉？ 2. 用一个兴奋性和一个抑制性神经元构造 Wilson-Cowan 网络模型，探讨参数与动力学行为关系。 3. 思考神经系统 persistent activity 的可能计算神经机制