数理解析研究所講究録
第 1544 巻 2007 年 163-167

163

Spontaneous Order of Self-organizing Systems*
Mau-Hsiang Shih\dagger and Feng-Sheng Tsai\ddagger

As Paul Adams wrote in the Joumal of Theoretical Biology in 1998 [1J, ‘Two of themost influential books in the history of biology are Darwin’s On the Origin of Sp.ecies .
and Hebb’s The Organization of,“ Donald Hebb’s classic book [6] is to have .
profound impact upon how populations of neurons interact collectively to perfom integrative activity underlying brain process. He postulated that the changes of synaptic$strengh_{8}$ based on coincidence detection could sustain persistent reverberatory activity in cortical circuits. The circulating neural impulses between populations of neurons
would continue to circulate, forming a diffuse self-assembling structure called “cell assemblies” [6, 13, 15].
There is increasing empirical support for Hebb’s contribution to neuropsychologicaltheory [3, 5, 11] and there also stimulates an intensive effort to promote the buildingof computer or network models of the brain based on Hebbian synaptic plasticity (thecoincidence-detection rule) [2, 4, 7, 9, 14, 16]. But there still lacks a complete mathematical explanation, a rigorous proof, that eonfirms the relation between Hebbian synaptic
plasticity and cell-assembly postulate.
The main problem concerning the relation between Hebbian synaptic plasticity andcell-assembly postulate can be formulated as follows:$Beha\tau\dot{n}or$$a$

The Consolidation Problem
Are there any organizing principles underlying the dynamical-combinatori process ofthe evolutionary network that allow one to describe neural populations underlying plasticity and to probe their time- and activity-dependent interactions capturing the characteristic property of the entire ensembles of cell-assemblies?
$al$

The model of the evolutionary network we are concerned with consists of a Populationof distinct inteyate-and-ffie Processing units (McCulloch-Pitts neurons or neurons) [10,12]; each constantly integrates all incoming signals transferred from synaPses on its cellbody and dendrites, and fires action Potentials to send signals to other neurons when–$suPsuPported$ by grants*Th18 work was suPsuPportedfrrants fromthe National Science Council of the RePublic of China and the Fbculty ofScience of National ?biwan Normal $Univer8ity$.
\dagger Department of Mathematics, National laiwan Normal University, 88 Sec. 4, Ting Chou Boad, 丁団 pei 116,$\tau u_{wan}$($m\epsilon hlh\alpha ath.$ ntnu. $edu.tw$ ).
$*D\epsilon partment$ of Mathematios,
National( $t\cdot t\iota a1hP716$ math. ntnu. edu. tw).

.

aiwan Normal University, 88 Sec. 4, Ting Chou Road, $IhIpei$ $116$ , Ilaiwan

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and denote by
the combined effort reaches its threshold. Name those $neuron81,$
the ordered pair $(i,j)$ the evolutionary coupling linking neuron to neuron . To eachneuron there is associated the threshold and the active state variable $x_{i}=0$ or 1,and to each evolutionary coupling $(i,j)$ there is associated the coupling strength variable. The phase space of the evolutionary network of coupled neurons is denoted by, the binary code consisting of all Ol-stringsof fixed-length . The
function hea is the Heaviside function: $hea(u)=1$ for $u\geq 0$ , otherwise . The nonlinear
dynamical system of the coupled neurons is then modeled by the following nonlinear
parametric equations:$\ldots,$

$n$

$j$

$i$

$i$

$b_{i}$$n$$a_{ij}$$\{0,1\}^{n}$$n$$x_{1}x_{2}\cdots x_{n}$

$0$

$n$

,

$t=0,1,$$x(t+1)$$=F_{s(t),t}(x(t))$$A(t+.1)$$=A(t)+D_{x(t)arrow x(t+1)}A$,$\ldots$,(1)$t=0,1,$$\ldots$,(2)

where $x(t)=(x_{1}(t), \ldots , x_{n}(t))$ is the neuronal active state at time $A(t)=(a_{ij}(t))_{n\cross n}$
is the evolutionary coupling state at time $s(t)$ denotes the neurons that adjust theiris the-state varying function encoding theactivity at time , anddynamics, whose ith component is defined by
$t,$

$t,$

$t$

$F_{\epsilon(t),t}(x)$$tim\triangleright \bm{t}d$$[F_{s(t),t}(x)]_{i}=x_{i}$if$i\not\in s(t)$,otherwise$[F_{\epsilon(t),t}(x)]_{i}=hea( \sum_{j=1}^{n}a_{ij}(t)x_{j}-b_{i})$.

of the parametric matrix
is called
The ith row and $jth$ column
the ultraderivative of the evolutionary coupling state at the evoIutionary coupling $(i,j)$ ,which varies with respect to the neuronal active states changing from $x(t)$ to $x(t+1)$ .
The ultraderivatives quantify plasticity of evolutionary couplings that allows the system
as a whole to undergo spontaneous organization.
of theAs the evolutionary network evolves, there are associated subsets
space $\{1, 2, \ldots, n\}\cross\{1,2, \ldots, n\}$ such that$\mathcal{D}_{i(t)arrow\alpha(t+1)}A$$\mathcal{D}oe(t)arrow x\langle t+1)a_{ij}$$\Lambda(t),$ $\Omega(t)$$\Lambda(t)$$=$$\{(i,j);\mathcal{D}ae(t)arrow x(t+1)\alpha_{j}>0\}$$\Omega(t)$$=$$\{(i,j);\mathcal{D}_{x(t)arrow\alpha(t+1)}a_{ij}\geq 0\}$,.

That raises the search for the algorithm that dictates the existence of $T\geq 0$ and asuch that the condition $1(x(t))=V$ holds true for all $t\geq T$ ,subsetof
where 1 $(x(t))$ denotes the collection of all active neurons at time , and that$V$$\{1, \ldots, n\}$$t$$\bigcap_{t\succeq T}\Omega(t)\supset VxV\supset.\bigcup_{t\geq T}\Lambda(t)$.

In this case, we say that the neurons in the subset are synchronized with $re8pect$ to$x(t)$ , accompanying a spontaneously organized distribution of the positive and negative
ultraderivatives to bind themselves into a single operational unit.
$V$

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The bewildering complexity with which the evolutionary network identified is inherently associated with two simple quantities to measure the driving forces of theevolutionary network’s dynamics. When a given neuron is active at time , there
are two classes emerging from the active evolutionary coupling strengths
for$i=1,$ $\ldots,n$ . The first consists of those with neurons being quiescent at time
but fired at $t+1$ , while the second consists of those with neurons being active
at time but quiescent at $t+1$ . By classifying all the active evolutionary couplingstrengths as the above fired-driven or unfired-driven character, we define the fired-driven
strength $FS(t)$ to be the sum of all $a_{ij}(t)s$ in the first class where is taken overneurons, whereas the unfired-driven strength US $(t)$ the sum of all $a_{1j}(t)s$ inall
the second class where is taken over all active neurons. The fired-driven strength$FS(t)$ is not necessarily greater than the unfired-driven strength US $(t)$ over time,
but when the discrete flow $x(t)$ behaves in the way that $x(t.)=x(t$ “ $\neq x(t)$ with(a feedback loop initiated by active neurons at time ), the fired-driven
emergestrength and the unfired-driven strength in the period oforderli$t$$j$$a_{1j}(t)$

$i$

$t$

$i$$t$$j$$act\dot{i}ve$$j$$)$$t_{*}\leq\hat{t}<t^{*}$$t_{*}$$t_{*}\bm{t}dt^{*}$$\cdot the$ness: $FS(t_{*})+FS(t_{*}+1)+\cdots+FS(t^{*}-1)>US(t_{*})+US(t_{*}+1)+\cdots+US(t^{*}-1)$.
This gives a fundamental law of pulsedynamics that combines dynamical and structuralcomplexity of the evolutionary network.
We.introduce what we call the “Hebbian evolving algorithm,” a generalized learning
rule analogous to a coincidence-detection rule of the Hebbian synaptic plasticity thatprovides a representative for the choices of the ultraderivatives in the growth dynamics
of the evolutionary network. They can be expressed in the formulas: $D_{x(t)arrow x(t+1):j}a\geq 0$
if neurons and are active simultaneously at time $t+1$ , otherwise $D_{x(t)arrow x(l+1)}a_{1j}<0$ ,and further we require that $|D_{x(t)arrow x(t+1)}a_{1j}|\geq|D_{x(t)arrow x(t+1)}a_{ji}|$ if $\delta_{ij}(t+1)>\delta_{ji}(t+1)$ ,$i$

$j$where $\delta_{ij}(t+1)=0$ or 1 is the indicator that indicates whether the active of neuron
at time has been able to change the active state of neuron at $t+1.-$
Defineto be $\sum_{i\in U}\min(\{a_{ii}(t);t=t_{*}, \ldots,t^{*}\})$ , a quantity that measuresthe minimal total excitability within the group of neurons
in the period of time
, where the coupling strength variableandwithis considered to be ameasure of excitability with respect to the neuron, according to the workingof the neuron , the increased excitability has a tendency to decrease the threshold forgenerating action potentials. Let us call that the discrete flow $x(t)$ generated by (1)iterates asynchronously if $s(t)$ is a singlton for all $t=0,1,$.
$j$

$t$

$i$

$\cdot$$E_{U}(t_{*},t^{*}.)$$U$$t^{*}$$t_{*}\leq t^{*}$$t$.

$a_{ii}$$i[8]\bm{t}d^{\backslash }$$i$$\ldots$

THEOREM. Consider the evolutionary network $ofn$ coupled neurons that subjects
to the dynamics (1) and obeys the Hebbian evolving algorithm. Given any initial neu, if $x(t)$ iterates asynchrvnously and theronal active state $x(O)$ in the phase space$\{0,1\}^{n}$166
minimal total excitability satisfies the assembling coordination:$E_{U}(t_{*}, t^{*}) \geq\sum_{i,j\in U}\max(\{a_{ij}(t)-a_{ji}(t)\geq 0;t=t_{*}, \ldots, t^{*}\}\cup\{0\})$for each non-empty subset
then there e.xist $T\geq$ .
$t\geq T$ and that

$0$

$U$

of

and for each $t_{*},t^{*}=0,1,$,with
of $\{1, 2, \ldots,n\}$ such that $1(x(t))=V$ for all$\{1, 2, \ldots, n\}$and a subset$V$(3)$\bigcap_{t\succeq T}\Omega(t)\supset VxV\supset\bigcup_{t\geq T}\Lambda(t)$$\ldots$$t_{*}\leq t^{*}$.

Consequ ently, starting with an initial neuronal active state $x(O)$ the Hebbian evolving
algorthm can be used to search a gmup of synchronized firing neurons, and the synchronized activity leads to a spontaneously organized distribution of the positive and negative
ultraderivatives that feed back to reinforce the neurons to fire in synchrvny, with positive
feedbacks conspiring to produce a cascade of sync-dependent neural circ uits, giving riseto the consolidation of neuml circuitry.

The above theorem is to formulate a consolidation problem of evolutionary networks concerning the relation between Hebbian synaptic plasticity and Hebbian cellassembly postulate, and to demonstrate a dominant theme of connectionist unravelinga dynamical-combinatorial process in a huge, interconnected system, in which the ongoing changes of the nodal-and.-coupling dynamics underlying plasticity are guaranteed
to result in group synchrony and syncsdependent circuits. Our proof of the theorem
shows a potential role of pulsedynamics in the stability of a nonlinear evolutionary networked system, concerning the “driving forces” derived from the evolutionary network’s
nodal and coupling activity, without invoking any physical “energy” to control system
dynamics.

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