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Neuromodulation of Signal-To-Noise (Associative Memory/Feedback/Locus Coeruleus/Norepinephrine) JAMES D

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Neuromodulation of Signal-To-Noise (Associative Memory/Feedback/Locus Coeruleus/Norepinephrine) JAMES D Proc. Natl. Acad. Sci. USA Vol. 86, pp. 1712-1716, March 1989 Neurobiology Noise in neural networks: Thresholds, hysteresis, and neuromodulation of signal-to-noise (associative memory/feedback/locus coeruleus/norepinephrine) JAMES D. KEELER*, ELGAR E. PICHLER, AND JOHN Rosst Chemistry Department, Stanford University, Stanford, CA 94305 Contributed by John Ross, December 1, 1988 ABSTRACT We study a neural-network model including this bifurcation a noise threshold. Second-order feedback in- Gaussian noise, higher-order neuronal interactions, and neu- teractions lead to hysteresis, multistability, a noise threshold romodulation. For a first-order network, there is a threshold at a higher noise level, and critical slowing down at the in the noise level (phase transition) above which the network thresholds. The signal-to-noise ratio can be adjusted in a bio- displays only disorganized behavior and critical slowing down logical neural network by neuromodulators such as norepi- near the noise threshold. The network can tolerate more noise nephrine. We suggest experiments to test hysteresis and if it has higher-order feedback interactions, which also lead to neuromodulation effects in biological networks and propose hysteresis and multistability in the network dynamics. The sig- a mechanism of norepinephrine in learning. nal-to-noise ratio can be adjusted in a biological neural net- work by neuromodulators such as norepinephrine. Compari- Phase Transition in a First-Order Neural Net with Noise sons are made to experimental results and further investiga- tions are suggested to test the effects of hysteresis and neuro- The McCulloch-Pitts (1) formal neuron is a binary unit modulation in pattern recognition and learning. We propose whose value depends on the linear sum of weighted inputs that norepinephrine may "quench" the neural patterns of ac- from the other neurons in the network. That is, at time t the tivity to enhance the ability to learn details. ith neuron receives an aggregate input, hi, from all of the other neurons given by Brain function depends on both the properties of individual n neurons and the neural network of which they are a part. hi(t) = IZ Tiju (, [1] Many abstract theoretical "neural network" models have n j=l been studied to understand how networks of interacting neu- rons give rise to emergent properties not displayed in single where uj is the output of thejth neuron, Tij is the interaction neurons (1-5). A vast amount of experimental evidence has strength between the ith and the jth neuron, Yi is a parame- been gathered regarding the neurophysiological, biochemi- ter representing the global strength of the synaptic connec- cal, and anatomical properties of individual neurons (6, 7). tions, and n is the number of neurons in the network. However, progress has been slow in incorporating these We assume that each neuron also receives noise input, (i, properties into abstract neural-network models (8), and along with the aggregate input, hi, from the other neurons. many open questions remain regarding the relationships be- Noise arises from many sources in a neuron (10), including tween the properties of individual neurons and the emergent spontaneous firing of the other neurons in the network. For properties of networks of neurons. simplicity, we approximate this noise, hi, as a random Gauss- The aim of the present investigation is to include three ex- ian variable with mean zero and standard deviation O'noise perimentally observed features of individual neurons into an (the results derived below are also valid for other, uncorre- abstract neural-network model and to investigate how these lated, noise distributions). The output of the ith neuron is properties are reflected in the emergent behavior of the neu- then taken to be the threshold of a sum of the weighted in- ral network: noise in a neuron, nonlinear (higher-order) in- puts plus this noise: teractions, and neuromodulation of signal-to-noise. We con- sider two issues regarding these additional features: (i) Do uN(t + 1) = g[hi(t) + (i(t)], i E 1, 2, 3, . ., n, [2] these modifications change the emergent behavior of the neural-network models? (ii) Are the changes relevant to ex- where the output function or gain function g(x) is a threshold perimental observations in biological networks? Affirmative function, g(x) = sgn(x), that has a value of +1 according to answers are obtained to both questions. the sign of the input. (We generalize g to a smooth function We begin by adding noise to the formal McCulloch-Pitts in the section on continuous equations.) (1) neuron and investigate how the behavior of a network of To analyze the behavior of this network, we project the such neurons changes as a function of the noise level. We current state of the neurons in the network, u(t), onto a par- use a projection-operator technique to reduce the stochastic ticular pattern p. Let y(t) denote this projection, dynamics of the network to a one-dimensional deterministic equation for the expected behavior of the system near a sta- = ble pattern. This technique is restricted to networks trained y(t) (1/n)p-u(t), [3] on random binary patterns, but it is general in the sense that so that y is a scalar that takes the value 1 when u = p, y 0 if it can be used on asymmetric networks, feed-forward net- u is some randomly chosen vector not correlated with p, and works, and temporal-sequence networks (9). The network y = -1 if u = -p. Hence, y is a measure of the correlation with only first-order interactions displays a bifurcation in the between u and network performance as the noise level is changed. We call p. Abbreviations: NE, norepinephrine; LC, locus coeruleus. The publication costs of this article were defrayed in part by page charge *Permanent address: Microelectronics and Computer Technology payment. This article must therefore be hereby marked "advertisement" Corporation, 3500 West Balcones Center Drive, Austin, TX 78759. in accordance with 18 U.S.C. §1734 solely to indicate this fact. tTo whom reprint requests should be addressed. 1712 Downloaded by guest on October 4, 2021 Neurobiology: Keeler et aL Proc. NatL Acad. Sci. USA 86 (1989) 1713 Using this definition for y, we can reduce the n equations tic equation for the expected behavior of y(t + 1) as a func- of Eq. 2 to an equation for y tion of y(t) (and, implicitly, the noise level): y(t + 1) = (1/n)p-g[h(t) + C(t)]. [4] (y(t + 1)) = 1 - 2p-[y(t)], [12] To proceed further, we must examine h. From Eq. 1 we where we have used p+ = 1 - p-. Since, by assumption, the know that h = y1T-u/n. To understand qualitatively how the patterns pa are randomly chosen, the distribution of 1L (cross- behavior of this network is affected by noise, examine the talk) is approximately Gaussian with standard deviation 0CTr behavior of the network in the vicinity of a fixed point pat- =-'y1(L - 1). Thus, we take iq to be a random Gaussian tern-i.e., g(Tp) = p. variable with zero mean ((7n) = 0) and variance o2 = As a concrete example, consider the autoassociative mod- OTnoise + (T2r (assuming independence between ; and 1L). el of Hopfield (5). Suppose that we have L patterns that we Given this assumption, we find that the probability ofgetting wish to store as fixed points: pa = (pg, p. ppg),, a = 1, 2, an error on the ith neuron is given by 3, . ., L, where pf is randomly chosen to be ± 1. Construct the connection matrix according to the Hebbian (11) learning rule: p y(t)] = I -x2/2dX [13] L lr ,[Y(t)] T = > [papa], [5] a=1 where r(y) is the signal-to-noise ratio, r(y) = yLy/a. The reduced Eq. 12 is a deterministic equation for the ex- with Ti, = 0 and where t denotes the transpose. Suppose that pected behavior of the system. If we start the system off we examine this system near the vicinity of one of the stored with correlation y(t) at time t and keep the noise level fixed patterns, u pa. Then the vector of aggregate inputs, h, can at a, Eqs. 12 and 13 tell us whether the expected correlation be written as at t + 1 increases or decreases as a function of the signal-to- L noise ratio, r[y(t)]. The boundary between the region where the + h = + X expected correlation, (y(t 1)), increases and the region yipO(p/3-u/n) y1/n papa.U [6] where it decreases is (y(t + 1)) = y(t)= yd-i.e., the fixed points of Eq. 12. Let y = (1/n)pp u, and write this last equation as To examine these fixed points as a function of a, we look for the intersections of (y) versus 1 - 2p-ir(y)]; the locus of intersections yields a plot of the fixed points (Fig. 1, curve h(t) = ylp1y(t) + ,u(t), [7] a). In this figure, we see that there is a threshold in the noise level, 0bth = YliV7 above which the only fixed point is y. where It) is the crosstalk that comes from all of the other = 0 (no pattern recognition). For a noise level below this stored patterns, threshold value, the network can have a nonzero fixed point that corresponds to partial recall of the pattern pa. The g = (t) P pau(t). [8] change from the ability to recognize a pattern (y. + 0) to n apt, cessation of such ability (y. = 0) can be viewed as a second- order phase transition in the signal-to-noise ratio, and it has Inserting Eq.
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