Supervised Learning in Neural Networks without Feedback Networks

In this paper we study the supervised learning in neural networks Unlike the com mon practice of back propagating error feedback by a separate feedback network that must have the same topology and synapse strengths as the feed forward network we propose a new adaptation algorithm by which the same supervised learning as accom plished by the back propagation algorithm can be achieved without using a separate feedback network The elimination of the feedback network makes it more likely for biological neural systems to achieve the same adaptation by means of some retrograde regulatory mechanisms that may exist in biological neural systems Another advantage of this new algorithm is that it simpli es hardware implementation of arti cial neural networks This research is supported in part by the National Science Foundation under grant ECS Feng Lin is with the Department of Electrical and Computer Engineering Wayne State University Detroit MI Tel Fax Email in ece eng wayne edu Robert D Brandt is with Intelligent Devices Inc Whittier Avenue Glen Ellyn IL He is currently visiting the Beckman Institute University of Illinois Urbana IL Version of April Introduction Since the back propagation algorithm for arti cial neural networks was proposed many years ago see Almeid Parker Pineda Piomelli et al Rumelhart et al Werbos Williams and other references researchers have speculated about whether an analogous adaptation mechanism might be observed in biological neural systems Zipser and Rumelhart The consensus among neuroscientists is that this is not likely Crick The main reason for this belief is that the back propagation is usually described as requiring a dedicated companion feedback network to propagate the error feedback see for example Hertz et al as illustrated in Figure This requirement of a separated feedback network is unlikely to be met in a biological neural system This is not to say that reciprocal connections are rare in biological neural systems in fact they are ubiquitous but rather that it is unlikely that a biological neural system could satisfy the strict requirement that there exists a one to one correspondence between connections in the feed forward and feedback networks and the corresponding connections in the two networks maintain identical strengths even as they adapt This seems even less likely given the fact that in most biological neural systems a connection between two neurons is composed of many even hundreds of synapses On the other hand neuroscientists have discovered several retrograde regulatory mecha nisms in biological neural systems such as modulation of adaptation by means of synaptic feedback via di usion of molecules such as nitric oxide and carbon monoxide Barinaga Schuman and Madison reuptake of neurotransmitter and neuromodulators De Camilli and Jahn Torri Tarelli et al action of second messengers Piomelli et al Schmajuk and DiCarlo Schmajuk and Blair retrograde axonal transport Trimble et al Vallee and Bloom and side e ects of synapse related RNA transcription The question is therefore how to explain this discrepancy in the back propagation algo rithm and what can actually happen in biological neural systems In particular whether a feedback network is absolutely necessary for adaptation to take place as suggested by the back propagation algorithm To put it another way can the error feedback or anything proportional to it be propagated without a feedback network but by some retrograde mech anism similar to those discovered by neuroscientists To obtain an a rmative answer to this question we focused on the simple observation that the relevant error feedback is implicit in the strengths of axonic synapses and their rates of change More precisely we show that the appropriate error feedback for any neuron is proportional to the derivative of the sum of the squares of the strengths of the axonic synapses Based on this observation we introduce a new adaptation algorithm that uses this implicit error feedback Using our algorithm there is no need to construct a separated feedback network The con guration of neural networks using our algorithm is therefore illustrated in Figure Here we use the word algorithm in a generalized sense to mean a mathematical description or model for generating error feedback and updating synapse strengths This property of our algorithm also makes the implementation especially the hardware implementation of arti cial neural networks much simpler than that has been previously considered for implementation of various adaptation algorithms see for example Lansner and Lehmann Linares Barranco et al Hollis and Paulos Dolenko and Card In particular the absence of the feedback network means that adding trainabil ity to a chip design does not involve additional wiring layout complexity between neurons A trainable neuron can be designed as a standard unit without considering network topology or adaptation coe cient These trainable neurons can then be connected in any way the de signer wants Obviously this increases the potential for designing networks with dynamically recon gurable topologies Our algorithm can be stated in either discrete time systems using di erence equations or continuous time systems using di erential equation In this paper we will present our algorithm for continuous time systems as biological neural systems generally run in a continuous asynchronous and concurrent fashion An algorithm for discrete time systems can be stated and proved in a similar manner using di erence equations Due to its similarity to the continuous time version we will not repeat it in this paper Preliminary results on our algorithm were presented in Brandt and Lin Brandt and Lin a Adaptation Algorithm In this section we present our main result a new adaptation algorithm for hierarchical and non hierarchical also called recurrent networks Because the networks considered in this paper can be interconnected arbitrarily standard notation based on layers are not very suitable so we use the following notations to specify the network con guration n is the label for a particular neuron s is the label for a particular synapse Dn is the set of dendritic synapses of neuron n An is the set of feedback generating axonic synapses of neuron n pres is the presynaptic neuron corresponding to synapse s and posts is the postsynaptic neuron corresponding to synapse s We use subscripts to associate variables with a particular synapse or neuron ws is the strength of synapse s pn is the membrane potential of neuron n and rn is the ring rate of neuron n A neuron transmits a signal from dendrites to axon This signal ow involves generation of postsynaptic potentials following activation of a synapse by neurotransmitters spatial and temporal integration of postsynaptic potentials at the soma triggering of action potentials In a biological neural system a connection between two neurons may consist several synapses Our algorithm applies to this general case However for an arti cial neural network one synapse for each connection is generally su cient In that case synapse is equivalent to connection along the axon hillock propagation of action potentials along the axon and the release of neurotransmitters from presynaptic terminals The ring rate rn of a neuron can be de ned as the reciprocal of the most recent interspike interval The synapse strength ws of a synapse is assumed to be proportional to the quantity of neurotransmitters released when a spike arrives at the synapse It is further assumed that the short time average of the postsynaptic potential is proportional to the product of the synapse strength ws and the presynaptic ring rate rpres The membrane potential at the axon hillock pn is thought to be a weighted sum of the postsynaptic potentials pn X