Mathematics For Human Nervous System
International Journal of Mathematics Trends and Technology- Volume27 Number1 – November 2015Mathematics for Human Nervous SystemMuhammad HanifAssociate Professor, Department of Applied MathematicsNoakhali Science and Technology UniversityNoakhali 3814, BangladeshAbstract: Modern digital computers outperformhumans in tasks based on precise and fast arithmeticoperations. However, people are much better andfaster than computers in solving complex perceptualproblems, such as recognizing images, often in thepresence of disturbances. Also, humans can performcomplex movements with precision and grace, evenin the presence of disturbances, and can generalizefrom past experience. In a computer, usually thereexists a single processor implementing a sequence ofarithmetic and logical operations, nowadays atspeeds approaching billion operations per second.However this type of devices has ability neither toadapt their structure nor to learn in the way thathuman being does. An Artificial Neural Model(ANM) is an information processing paradigm that isinspired by the way biological nervous systems, suchas the brain, process information. The key element ofthis paradigm is the novel mathematical structure ofthe information processing system. In this paper wefocus on the mathematical modeling aspects of thebasic unit of the human nervous system and ArtificialNeural Model.Figure 1 The Human Nervous SystemThe central nervous system is so named because ofits anatomical location along the central axis of thebody and because it is central in function. If we use acomputer analogy to understand that it is central infunction, the CNS would be the central processingunit and the other two parts of the nervous systemwould supply inputs and transmit outputs. Figure 3.2shows the central nervous system.a. Major Subdivisions of the Central NervousSystem. The major subdivisions of the centralnervous system are the brain and spinal cord.Keywords: Human Nervous System (CNS), ArtificialNeural Model (ANM), Basic Mathematics.I THE HUMAN NERVIOUS SYSTEMThe human nervous system is divided into two majordivisions: The peripheral nervous system (PNS) andthe central nervous system (CNS). The PNS consistsof a. sensory neurons running from stimulusreceptors that inform the CNS of the stimuli. b.Motor neurons running from the CNS to the musclesand glands - called effectors - that take action. Theperipheral nervous system is subdivided into the a.sensory-somatic nervous system and the autonomicnervous system (ANS). The peripheral nervoussystem carry information to and from the centralnervous system. The central nervous system iscomposed of the brain and spinal cord.ISSN: 2231-5373Figure 2 The central nervous system (CNS).b. Coverings of the Central Nervous System. Boneand fibrous tissues cover the parts of the centralnervous system. These coverings help to protect thedelicate tissue of the CNS.http://www.ijmttjournal.orgPage 12
International Journal of Mathematics Trends and Technology- Volume27 Number1 – November 2015c. Cerebrospinal Fluid. The cerebrospinal fluid (CSF)is a liquid that is thought to serve as a cushion andcirculatory vehicle within the central nervous system.A neuron is a nerve cell body and all of its branches.II THE HUMAN BRAINThe human brain is one of the most complex objectsin the universe. Many attempts have been made toinvestigate and model the functionalities of the brain.We still do not know how exactly brain learns.However, the human brain has three majorsubdivisions: brainstem, cerebellum, and thecerebrum. The central nervous system is first formedas a simple tube like structure in the embryo. Theconcentration of nervous tissues at one end of thehuman embryo to produce the brain and head isreferred to as cephalization. When the embryo isabout four weeks old, it is possible to identify theearly forms of the brainstem, cerebellum, and thecerebrum, as well as the spinal cord. As developmentcontinues, the brain is located within the cranium inthe cranial cavity. Figure 3.3 for illustrations of theadult brain.Figure 4 A neuronFigure 3 Human brain: A side view, B bottom view.III Neuron and Neuron “Connections”b. Dendrite. The dendrite is a neuron process which isa tree-like structure that receives the signals fromother neurons and carries the signals toward the cellbody. Each neuron may have one or more dendrites.Dendrites receive information and transmit it to thecell body.c. Axon. The axon is a single long fiber that carriesthe signal from the cell body out to other neurons.Each neuron has only one axon. An axon, having alength varying from a fraction of a millimeter to ameter in human body, prolongs from the cell body atthe point called axon hillock. At the other end, theaxon is separated into several branches, at the veryend of which the axon enlarges and forms terminalbuttons.d. Synapses. The terminal buttons of an axon areplaced in special structures called the synapses whichare the junctions transmitting signals from oneISSN: 2231-5373Figure 5 Soma, Dendrite, Axon and Synapse of aNeuron.There are four types of neuron branches: The cellbody, dendrites, axons and synapsis.a. The Cell Body. A neuron has a roughly sphericalcell body called soma. The cell body is the heart ofthe cell, containing the nucleus and maintainingprotein synthesis. The cell body is fully responsiblefor growth and maintenance of the neuron.neuron to another . A neuron typically drive 10 3 to104 synaptic junctions. Although it is not verycommon, synapses may also take place between twoaxons or two dendrites of different cells or betweenan axon and a cell body.The human central nervous system(CNS) iscomprised of about 1.3x1010 neurons and that about1x1010 of them takes place in the brain i.e. the brainhas 1010 100 billion neurons.The thickness of a banknote is approx. 0.1 mm, i.e., the stack of 100 billionbank notes has the length of 100 km. Each neuron has104 connections to other neurons in the brain or spinalcord. i.e, the network is sparsely connected. At anytime, some of these neurons are firing and the powerdissipation due this electrical activity is estimated tobe in the order of 10 watts. Monitoring the activity inthe brain has shown that, even when asleep, 5x10 7nerve impulses per second are being relayed back andhttp://www.ijmttjournal.orgPage 13
International Journal of Mathematics Trends and Technology- Volume27 Number1 – November 2015forth between the brain and other parts of the body.This rate is increased significantly when awake.Electrical signals travel along these connections, andeach neuron processes its inputs and generates a setof output signals which are then sent to neurons thatit is connected to.A neuron may "connect" either with another neuronor with a muscle fiber. A phrase used to describesuch "connections" is "continuity without contact."Neurons do not actually touch. There is just enoughspace to prevent the electrical transmission fromcrossing from the first neuron to the next. This spaceis called the synaptic cleft. Information is transferredacross the synaptic cleft by chemicals ctured and stored on only one side of the cleft.Because of this, information flows in only onedirection across the cleft.We conclude the biological neuron model with thefollowing six points:1. A neuron has a single output conducted throughits axon and multiple inputs conducted throughdendrites.Though the nerve cell frequently possesses more thanone dendrite,the axon is single The dendrite is the receptiveprocess of the neuron;the axon is the discharging process, .2. The single neuronal output, produced by a pulseknown as the action potential, can be fanned out toconnect to many other neurons. Signal flow isunilateral at junctions.An axon gives rise to many expanded terminalbranches ( presynaptic terminal boutons) A single neuron may be involved in many thousandsof synaptic connections,but in every case the impulse transmission canoccur only in one direction .3. The dendrites ( and a neuron may have as many as150,000 of them) are the major source of inputsalthough it is noted that inputs can also be madethrough the soma ( body) and axon and the neuron asa whole can be responsive to fields in itsenvironment. Hence, the model has multiple inputs,and there are many of them.[excitatory postsynaptic potential]. TheEPSPs willthe sum, and if thecritical level of depolarization is reached, an impulsewill be set off.5. The neuronal response is all or none in that thecharacteristics of the pulse (referred to as the actionpotential) do not depend on detailed characteristicsof the input.Thus, the propagated disturbance established in asingle nerve fibercannot be varied by grading the intensity or durationof the stimulus,i.e., the nerve fiber under a given set of conditionsgives a maximalresponse or no response at all.Hence, our model should exhibit theresholding and alimited range set of possible outputs. The single pulsecase is that there are only two possible outputs with“1” representing the action potential and “0” or “-1”representing the quiescent state.6. The mechanisms for changing the response of aneural network include the use of chemicals likecalcium and neurotransmitters to change thecharacteristics of a neuron (open and close selectedion channels) as well as changing the synapticstrengths. Although only the second possibility isexploited in artificial neural networks but the firstmechanism is the most important.4. Artificial Neuron Model and BasicMathematicsArtificial neuron model proposed by McCulloch andPitts, [McCulloch and Pitts 1943] attempt toreproduce. An artificial neuron is an informationprocessing unit that is fundamental to the operationof a neural network. Each neuron represents a map,typically with multiple inputs and a single output.Specifically, the output of the neuron is a function ofa sum of the inputs. The function at the output of theneuron is called the activation function.4. The neuronal response depends upon asummation of inputs.The two subliminal volleys are sent in over the samenerve, each volleyproduces an effect which is manifested by an EPSPISSN: 2231-5373http://www.ijmttjournal.orgPage 14
International Journal of Mathematics Trends and Technology- Volume27 Number1 – November 2015Figure 2 Activity similarities between (a) acomputing neuron and (b) a biological neuron(a)(b)Figure 1 (a), (b) Mathematical representation of anartificial neuron modelSingle-Input Artificial Neuron ModelA single-input artificial neuron is shown in Figure 2.The scalar input x is multiplied by the scalar weightw to form wx, one of the terms that is sent to thesummer. The other input, 1, is multiplied by a bias band then passed to the summer. The summer outputn, often referred to as the net input, goes into aactivation function f, which produces the scalarneuron output y.If we relate this simple model back to the biologicalneuron that the input x is a single dendrite, the weightw corresponds to the strength of a synapse, the cellbody is represented by the summation and theactivation function, and the neuron output yrepresents the signal on the axon;ISSN: 2231-5373Figure 3 Single-Input NeuronThen the neuron output is calculated asy f ( wx b).If, for instance, w 3, x 2 and b -1.5, theny f (3(2)-1.5) f(4.5)The actual output depends on the particular activationfunction that is chosen.The bias is much like a weight, except that it has aconstant input of 1. One can choose neurons with orwithout biases. The bias gives the network an extravariable, and so we might expect that networks withbiases would be more powerful than those without,and that is true. Note, for instance, that y neuronwithout a bias will always have a net input n of zerowhen the network inputs x are zero. This may not bedesirable and can be avoided by the use of a bias. Infact, w and b are both adjustable scalar parameters ofthe neuron. Typically the activation function ischosen by the designer and then the parameters w andb will be adjusted by some learning rule so that theneuron input/output relationship meets some specificgoal.http://www.ijmttjournal.orgPage 15
International Journal of Mathematics Trends and Technology- Volume27 Number1 – November 2015As described in the following section, we havedifferent activation functions for different purposes.5 Activation FunctionsA particular activation function is chosen to satisfysome specification of the problem that the neuron isattempting to solve. A variety of activation functionshave been included in this section. Three of the mostcommonly used functions are discussed below. Thehard limit activation function, shown on the left sideof Figure 4, sets the output of the neuron to 0 if thefunction argument is less than 0, or 1 if its argumentis greater than or equal to 0. We will use this functionto create neurons that classify inputs into two distinctcategories.Figure 5 Linear Activation FunctionThe log-sigmoid activation function is shown inFigure 6.y logsig(n)y logsig(wx b)Log-Sigmoid Activation FunctionSingle-Input logsig NeuronFigure 6 . Log-Sigmoid Activation FunctionThis activation function takes the input (which mayhave any value between plus and minus infinity) andsquashes the output into the range 0 to 1, according tothe expression:y hardlim(n)y hardlim(wx b)Hard Limit activation FunctionSingle-Input hardlim NeuronFigure 4 Hard Limit Activation FunctionThe graph on the right side of Figure 4 illustrates theinput/output characteristic of a single-input neuronthat uses a hard limit activation function. Here wecan see the effect of the weight and the bias. Notethat an icon for the hard limit activation function isshown between the two figures. Such icons willreplace the general f in network diagrams to show theparticular activation function that is being used.The output of a linear activation function is equal toits input:y nas illustrated in Figure 5.Neurons with this activation function are used in theADALINE networks.y purelin(n)y purelin(wx b)Linear Activation FunctionSingle-Input purelin NeuronISSN: 2231-5373y11 enThe log-sigmoid activation function is commonlyused in multilayer networks that are trained using thebackpropagation algorithm, in part because thisfunction is differentiable. Most of the activationfunctions used in the artificial neural networks aresummarized in Table Page 16
International Journal of Mathematics Trends and Technology- Volume27 Number1 – November nearposinCompetitivecompetxiefficacy wiinputs synapticweightsexcitation level b noise(bias,threshold)process inputfactivation function axonsignalynetwork output(a)Table 1 Activation FunctionsMultiple-Input Artificial Neuron ModelTypically, a neuron has more than one input. Aneuron with inputs x1 , x 2 ,., x N is shown inFigure 7. The individual inputs are each weighted bycorresponding elements w11 , w12 ,., w1N of theweight matrix W.(b)Figure 8 (a) Comparison between the biological andartificial neuron(b) Structure of artificial neuron model frombiological modelAcknowledgementFigure 7 Comparison between (a) a artificial neuronand (b) a biological neuronThe University Grants Commission Bangladesh forproviding the financial support to work with thisarticle.References[1] Muhammad Hanif, „‟ Unconstrained Nonlinear OptimizationMethods for Feedforward Artficial Neural Networks”, PostDoctoral Research Report , University Grants CommissionBangladesh (2015), pp. 120-179[2] Muhammad Hanif and Syeda Latifa Parveen, “Newton‟sMethod for Feedforward Artificial Neural Networks”, USTA(2014), V- 21, Number 1, pp. 59-67.ISSN: 2231-5373http://www.ijmttjournal.orgPage 17
International Journal of Mathematics Trends and Technology- Volume27 Number1 – November 2015[3] Muhammad Hanif, Md Jashim Uddin, Md Abdul Alim „‟ TheMethod of Steepest Descent for Feedforward Artificial NeuralNetworks”, IOSR Journals(2014), V- 10, Issue 1, pp. 53-61[4] Muhammad Hanif and Jamal Nazrul Islam „‟ First-OrderOptimization Method for Single and Multiple-Layer FeedforwardArtificial Neural Networks”, Journal of the BangladeshElectronics Society (2010), V- 10, Number 1-2, pp. 33-47.[5] Carpenter, G. and Grossberg, S. (1998), The Handbook ofBrain Theory and Neural Networks, (ed. M.A. Arbib), MIT Press,Cambridge, MA, pp.79–82.[6] X. H. Hu and G. A. Chen (1997), “Efficient backpropagationlearning using optimal learning rate and momentum", NeuralNetworks 10(3), 517-527.[7] P. Mehra and B. W. Wah, “Artificial neural networks: conceptsand theory,” IEEE Comput. Society Press, (1992).[11] R. Battiti (1992), “First and second-order methods forlearning: Between steepest descent and Newton‟s methods,"Neural Computation 4, 141-166.[12] J.S. Judd, “Neural Network Design and the complexity ofLearning,” MIT Press, Cambridge, MA (1990).[13] P.R. Adby and M.A.H. Dempster, “Introduction toOptimization Methods,” Haisted Press, New York (1974).[14] McCulloch, W. and W. Pitts (1943), A Logical Calculus ofthe Ideas Immanent in Nervous Activity, Bulletin of MathematicalBiophysics, 5, 115-133.[15] Ankit Anand, Shruti, Kunwar Ambarish Singh (2014), AnEfficient Approach for Steiner Tree Problem by GeneticAlgorithm, SSRG International Journal of Computer Science andEngineering , volume2 issue5 , 233-240.[16] Jinesh K. Kamdar(2014), Human Computer Interaction - AReview , SSRG International Journal of Computer Science andEngineering , volume1 issue7, 27-30BiographyMuhammad Hanif received his B.Scand M.Sc degree in MathematicsfromChittagongUniversity,Chittagong, Bangladesh in 1996 and1998 respectively and M.Phil andPhD degree in Applied MathematicsfromResearchCentreforMathematical and Physical Sciences,University of Engineering and Technology. In 2006,he joinedNoakhali Science and TechnologyUniversity, Noakhali, Bangladesh where he iscurrently an Associate Professor in the department ofApplied Mathematics. His current research interestsinclude number theory, nonlinear programming,biomedical mathematics, neural networks andindustrial mathematics.Chittagong University, Bangladesh in 2007 and 2012respectively under the renowned cosmologist Professor J NIslam (1939 – 2013). He did Post Doctoral Research inApplied Mathematics from University Grants CommissionBangladesh in 2014. From 2000 to 2006 he worked as aLecturer and Assistant Professor in Mathematics atInternational Islamic University Chittagong and ChittagongISSN: 2231-5373http://www.ijmttjournal.orgPage 18
peripheral nervous system is subdivided into the a. sensory-somatic nervous system and the autonomic nervous system (ANS). The peripheral nervous system carry information to and from the central nervous system. The central nervous system is composed of the brain and spinal cord. Figure 1 The Human Nervous System
I. Central Nervous System vs Peripheral Nervous System II. Peripheral Nervous System A. Somatic Nervous System B. Autonomic Nervous System III. Autonomic Nervous System A. Parasympathetic Nervous System B. Sympathetic Nervous System IV. Reflex Actions V. Central Nervous Sys
nervous system is separated into two major divisions: the central nervous system and the peripheral nervous system. The central nervous system is the control center of the body. The functions of the central nervous system are similar to those of the central processing unit of a computer. The central nervous system relays messages, processes .
Nervous System: Consists of two main divisions: Central nervous system (CNS) Brain and spinal cord Peripheral nervous system: Somatic nervous system Autonomic nervous system Central Nervous System
Divisions of the nervous system 1. Central Nervous System (CNS) 2. Peripheral Nervous System (PNS) Central Nervous System (CNS) a. Brain b. Spinal cord . Peripheral Nervous System (PNS) a. Sensory b. Motor . Divisions of the Nervous System which consists of is divided into that make up which is divided into The Nervous
The nervous system can be divided into two major regions: the central and peripheral nervous systems. The central nervous system (CNS) is the brain and spinal cord, and the peripheral nervous system . function of the nervous system. oT describe the functional divisions of the nervous system, it is important
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the nervous system controls the complex activities of the human body. Overview of the Nervous System We will begin our tour of the nervous system by taking a broad look at its two major divisions, the central nervous system (CNS) and the peripheral nervous system (PNS). Even though these parts work together as a highly efficient, integrated unit,
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