Section II is self-contained mathematical reasoning on the extended communication system, and its application to neural firing and connections will be left to Section III , where the settings, definitions, and arguments in Section II are interpreted accordingly. In Section IV , retrospectively we argue that, our communication system and its mathematics can be applied to thermodynamic system.

Back inan equilibrium ensemble at a macroscopic level of the actual system allows us to investigate other macroscopicFor example, in a closed isothermal system given specific volume and temperature, its so-called canonical ensemble can be used to calculate the average value of energy or pressure, based on the fundamental postulate: time average of a macroscopic variable in thermodynamic system is equal to ensemble average of the variable. We again presume that the fundamental postulate holds true in the case of neural firing in equilibrium. That is to say, if we could properly take samples from thesystem in equilibrium repeatedly over time (imagine that a magical camera could take snapshots ofand reveal their firing), theof all possible samples, each identified by the firing situation ofwould be consistent with the equilibrium ensemble. As with microstate, the samples are supposed to be under the probabilistic intervention of neural connections. Then, we can use the sample sequence in temporal order as the subject to investigate the existence and unavoidability of neural firing equilibrium. This approach has its advantages compared to using the ensemble. First, the sample sequence is supposed to be less impractical than the imagery ensemble. Second, with the concepts such as sequence of successive samples,of samples, and intervention by connections, we have necessary ingredients to compose a communication system with a noisy discrete channel, so as to proceed with our investigation in the framework of well-knownMeanwhile, to mimic the influence of neural firing on neural connections, the conceptualized communication system, is extended such that the discrete noisy channel are not fixed, but dependent of and changeable by received signal. We will show that, it is inevitable for the channel to be attracted into fixed point, implying that, neural connections are to be attracted into fixed propagation probabilities, and thus neural firing is to evolve to new equilibrium.

Our research suggests so. We will show that, the new equilibrium ensemble is not only possible but also inevitable even if thesystem has enormous connections in it. As our main conclusion, it follows immediately that, under dynamically changeable connections, the equilibrium of neural firing is evitable. We notice that, some notions in ourhave been employed by known neuralmodels. For example, Hopfieldor its variant Boltzmann machine,inspired by Ising modelin physic, proposed aof simple binary units, where any repeated updating the state (+1 or −1) of a randomly chosen unit will cause theto converge to the same combination of units’ state. Although not fully comparable, our model stresses the probabilisticof neural connections in propagatingand its conclusions, as it turns out, are less remote from the reality.

Following the presumption, we can notice that neural connections can intervene in the equilibrium ensemble of neural firing, making it evolve. Suppose that a neural connection is established between two otherwise disconnectedsay, from presynaptic1 to postsynaptic2. If the connection is in a perfect condition, everyfrom1 will propagate to2 and then trigger2 to fire, which means that1 and2 will always fire together if the duration of propagation is sufficiently short to be ignored. Consequently, the equilibrium ensemble instantaneously jumps to a new one, where the microstates, with1 but2 not firing, vanish in the new equilibrium ensemble, giving up their probabilities to the microstate with1 and2 firing together. Neural connection, however, is not always a steady unblocked path to transmitOn one hand, neural connections vary in their ability to propagateIf our connection is too weak to propagate everyfrom1 to2 successively, not every firing of1 will trigger2 to fire. In that case, new equilibrium ensemble is possible only if our connection propagatesstochastically with fixed success rate. Assuming so,a connection’s ability to propagate anto postsynapticcan be measured by probability. On the other hand, the ability of connection is to strengthen or weaken over time, in response to increases or decreases in stimulations, specifically, the propagation ofover the connection.To be illustrative,that “fire together, wire together” as summarized from HebbianIn that case, for new ensemble to exist, our connection must develop to attain fixed propagation probability. Is that possible?

Inan ensemble is a mental collection of a large number of virtual systems, each constructed to be a replica of the actual system. In other words, an ensemble is aover all possible microstates of the system. We adopt the concepts of ensemble and microstate to the realm ofsystem, which, like thermodynamic system, is an almost homogeneous collection of massive individuals. Asystem consisting ofcan be considered to have at most countable 2microstates, if a microstate is to represent one particular collective situation ofneurons’ binary conditions, firing or not firing. And, because the ensemble for the thermodynamic system in equilibrium doesn’t evolve over time, we accordingly presume that, for asystem, an equilibrium condition with respect to neural firing is possible, where the ensemble is independent of time. In the absence of neural connections, neural firing are triggered by the causes such as the external stimuli from surrounding environments directed to sensoryor the self-excitability associated with neurons’ intrinsic electrical fluctuations.When environment provides stimuli and self-exciting occurs stochastically, thesystem without connections can be considered to be in equilibrium condition.

an electrically excitable cell, usually many in a collaborative way, is the driving force of lives’ behaviors and thoughts. Responding to touch, sound, light and all other stimuli,“fire”a short-lasting event in which the electrical membrane potential ofrapidly rises and falls.is connected to many otherby developing axon and dendrites tree, branches of which make synaptic contact with otherfrom atravels down the one-way connections and triggers otherto firewhich likewise trigger further downstreamto fire, so on and on. So, with neural connectionsfrom onecan propagate to many others. In this paper, we propose a mathematicalregarding the dynamics in neural firing, inspired by

II. AN EXTENDED COMMUNICATION SYSTEM Section: Choose Top of page ABSTRACT I.INTRODUCTION II.AN EXTENDED COMMUNICAT... << III.APPLICATION TO NEURAL... IV.DISCUSSIONS REFERENCES CITING ARTICLES

1 We shall build up the communication system as schematically indicated in Figure, represent the concepts involved as mathematical entities, and then look into their relation.

information source produces sequence of successive symbols at a frequency of f symbols per unit time. The source can produce n possible different symbols, each of which is represented by S i or S j where 1 ≤ i, j ≤ n. And the source can be represented by an ergodic Markov process, from which roughly all sufficient long sequences produced have the same probability distribution of symbols. Let the probability of symbol S i in the sequences be represented by p(i) ∈ [0, 1]. Then the sequences produced by the source can be described by n symbols’ probabilities. We write I to denote such n-turple of symbol probabilities. I = p ( 1 ) , p ( 2 ) , p ( 3 ) , … , p ( n ) (1) The discretesource produces sequence of successive symbols at a frequency ofsymbols per unit time. The source can producepossible different symbols, each of which is represented byorwhere 1 ≤. And the source can be represented by an ergodic Markov process, from which roughly all sufficient long sequences produced have the sameof symbols. Let the probability of symbolin the sequences be represented by) ∈ [0, 1]. Then the sequences produced by the source can be described bysymbols’ probabilities. We writeto denote such-turple of symbol probabilities.

n2 inter-symbol transition probabilities p i (j) ∈ [0, 1], the probability of transmitted symbol S i being received by destination as S j . We write e to denote such n2-turple of transition probabilities. All possible e form a compact and convex set of n2-dimensional cube. e = p 1 ( 1 ) , p 1 ( 2 ) , p 1 ( 3 ) , … , p 1 ( n ) , p 2 ( 1 ) , p 2 ( 2 ) , p 2 ( 3 ) , … , p 2 ( n ) , ⋮ p n ( 1 ) , p n ( 2 ) , p n ( 3 ) , … , p n ( n ) (2) The successive symbols are independently perturbed by the noisy such that, the channel can be described by the set ofinter-symbol transition probabilities) ∈ [0, 1], the probability of transmitted symbolbeing received by destination as. We writeto denote such-turple of transition probabilities. All possibleform a compact and convex set of-dimensional cube.

probability distribution of received symbols can also be described by an n-turple of symbol probabilities, which we denote by O. O = p ′ ( 1 ) , p ′ ( 2 ) , p ′ ( 3 ) , … , p ′ ( n ) (3) The destination receives the symbol sequences perturbed. Theof received symbols can also be described by an-turple of symbol probabilities, which we denote by

O is determined by I and e. For any entry p′(j) of O, p ′ ( j ) = ∑ i = 1 n p ( i , j ) = ∑ i = 1 n p ( i ) p i ( j ) is determined byand. For any entry′() of

I, e) to O, denoted by φ. φ is continuous on e by its definition. O = φ ( I , e ) (4) Accordingly, there exists a mapping from () to, denoted byis continuous onby its definition.

n2 continuous mappings from O to p i (j), each of which is denoted by ϕ ij . p i ( j ) = ϕ ij ( O ) (5) Now we assume that our noisy discrete channel depends on received signal, i.e., the received sequence of symbols. For our purpose, the dependence is assumed to be described by a collection ofcontinuous mappings fromto), each of which is denoted by

n2 such θ ij , all of which can be merged into one single continuous mapping ϕ. e = ϕ ( O ) (6) There aresuch, all of which can be merged into one single continuous mapping

p i (j) in e satisfies Here any entry) insatisfies (5)

I as information source, is possible. Then there exists an e such that O = φ(I, e) and e = ϕ(O) according to Eq. e = ϕ(φ(I, e)), the right side of which gives rise a function θ with respect to e. θ is a composition of φ and ϕ, and thus continuous on e. θ ( e ) = ϕ ( φ ( I , e ) ) (7) Suppose that, under that assumption, a communication system, givenassource, is possible. Then there exists ansuch that) and) according to Eq. (4) and Eq. (6) . So)), the right side of which gives rise a functionwith respect tois a composition ofand, and thus continuous on

theorem, 13 Über abbildungen von mannigfaltigkeiten ,” Mathematische Annalen 71, 97 (1911). 13. L. E. J. Brouwer, “,” Mathematische Annalen, 97 (1911). https://doi.org/10.1007/BF01456931 θ is a continuous function on e and all possible e form a compact and convex set, there must be a fixed point e fp such that e fp = θ(e fp ). Therefore, the existence of our communication system requires only the assumption of ϕ’s continuity with respect to O. θ might have more than one fixed point, while the rest of e, of course, are nonfixed points because θ(e)≠e. According to Brouwer’s fixed-pointifis a continuous function onand all possibleform a compact and convex set, there must be a fixed pointsuch that). Therefore, the existence of our communication system requires only the assumption of’s continuity with respect tomight have more than one fixed point, while the rest of, of course, are nonfixed points because)≠

probabilities distribution O, the transition probability from symbol S i to S j would develop eventually to p i (j). And, likewise Eq. I, if the channel stays fixed at e, the received sequences would be fixed at O = φ(I, e), and under O the channel would change towards e ′ = ϕ O = θ ( e ) unless e is already a fixed point such that e′ = e. There is contradiction here since the channel was assumed to be fixed at e. Here, we shall treat θ(e) as the instantaneous tendency of the channel standing at e. e′, as the destination of changing channel, wouldn’t be reached; its value, alone, is trivial for our purpose. However, by comparing each entry p′(j) in e′ to its corresponding entry p i (j) in e, we know how the channel changes in the direction of each inter-symbol transition probability. Specifically, the transition probability from S i to S j is to increase if p i ′ ( j ) > p i ( j ) , decrease if p i ′ ( j ) < p i ( j ) , or remain if p i ′ ( j ) = p i ( j ) . Therefore, we can talk about tendency for individual transition probability. According to Eq. i and j we can designate one function θ ij with p i (j) as variable. θ ij p = ϕ ij φ I , p ; e (8) Now we further assume that the channel is changeable over time by received signal. In that case, we retranslate Eq. (6) by stating that, if the received sequences, somehow, could be managed to have fixed symbol, the transition probability from symboltowould develop eventually to). And, likewise Eq. (7) states that, given, if the channel stays fixed at, the received sequences would be fixed at), and underthe channel would change towardsunlessis already a fixed point such that′ =. There is contradiction here since the channel was assumed to be fixed at. Here, we shall treat) as the instantaneous tendency of the channel standing at′, as the destination of changing channel, wouldn’t be reached; its value, alone, is trivial for our purpose. However, by comparing each entry′() in′ to its corresponding entry) in, we know how the channel changes in the direction of each inter-symbol transition probability. Specifically, the transition probability fromtois to increase if, decrease if, or remain if. Therefore, we can talk about tendency for individual transition probability. According to Eq. (4) and Eq. (5) , for anyandwe can designate one functionwith) as variable.

p is short for variable p i (j) for convenience, and (p; e) represents a specific e with the particular p i (j) entry substituted by variable p. Thus, given I and e, θ ij p indicates the tendency of p i (j) standing at (p; e), by assuming that the other entries of e are fixed. Figure 2 p i (j) is always expelled from the nonfixed point and attracted into the nearest fixed point. Hereis short for variable) for convenience, and () represents a specificwith the particular) entry substituted by variable. Thus, givenandindicates the tendency of) standing at (), by assuming that the other entries ofare fixed. Figureillustrates that,) is always expelled from the nonfixed point and attracted into the nearest fixed point.

Now there are n2 such θ ij (p) functions, and correspondingly n2 such p i (j) variables are changing towards their own fixed points at the same time. Each θ ij ’s fixed point could keep shifting because other p i (j) aren’t fixed. Consequently, it might take a longer time for each p i (j) to finally reach its fixed point. To conclude, the transition probabilities of our channel are always to be attracted into fixed points, although the trajectory of their changing, especially for large n, might be overwhelmingly untraceable and long-lasting. It is essential to note that, the probability distribution of received symbol sequence, represented by I, will also settle down on fixed value as soon as the channel reaches its fixed point.

We can see that the tendency function θ(e) plays a key role in our reasoning regarding the fixed point of channel. The rest of this section gives some other nontrivial conclusions.

Again, e is fixed point as long as θ(e) = ϕ(φ(I, e)) = e. For an otherwise fixed point e to nonfixed, we must make sure θ(e)≠e, for example, by altering I or the mapping ϕ. So, we can alter our communication system to deviate the channel from fixed point from time to time, and the channel, with resilience, can always finds its way back. If θ has more than one fixed point, the system could evolve into any one of them, depending on the nonfixed point e nfp to start with. As a special case, if there exists one single fixed point for θ, the initial e nfp is not relevant anymore since given any e nfp there is to be the sole e fp .