Simulating Human Single Motor Units using Self-Organizing Agents

Onder Gurcan∗†, Carole Bernon†, Kemal S. Turker‡, Jean-Pierre Mano§, Pierre Glize† and Oguz Dikenelli∗∗Ege University, Computer Engineering Department, Izmir, Turkey

†Paul Sabatier University, Institut de Recherche en Informatique de Toulouse, Toulouse, France‡Koc University, School of Medicine, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey

§Upetec, Ramonville St Agne, Toulouse, France

Abstract—Understanding functional synaptic connectivity ofhuman central nervous system is one of the holy grails of theneuroscience. Due to the complexity of nervous system, it iscommon to reduce the problem to smaller networks such asmotor unit pathways. In this sense, we designed and developeda simulation model that learns acting in the same way ofhuman single motor units by using findings on human subjects.The developed model is based on self-organizing agents whosenominal and cooperative behaviors are based on the currentknowledge on biological neural networks. The results show thatthe simulation model generates similar functionality with theobserved data.

Keywords-self-wiring; biological neural networks

I. INTRODUCTION

Understanding the architecture of brain and mind is

one of the grand challenges accepted by the UK Com-

puting Research Committee (UKCRC) so far [1]. Despite

its importance, our understanding on how these highly-

complex highly-parallel interconnected systems of neurons

and synapses work is very basic. However, knowledge of the

synaptic connections between neurons is a key prerequisite

to understand the operation of the nervous system. While

the anatomy of these connections can be obtained by histo-

chemical methods, their functional connections can only be

determined using electrophysiological recordings. Compared

with the studies on human subjects, interpretations of the

connections of the activated fibers and neurons are straight-

forward and easy in animal experiments since it is possible

to make direct measurements. Although direct experiments

cannot be performed in humans, various indirect experiments

are performed to explore functional connections. However,

there are still gaps in our understanding of these wirings due

to technical difficulties. In addition, there is no satisfactory

theory on how these unknown parts of central nervous

system (CNS) operate. Therefore, neuroscientists still rely

upon the knowledge that is obtained in animal studies.

Thus, human studies revealing functional connectivity at the

network level are still missing.

CNS can be understood in terms of complex networks.

Characterizing structure and function of complex networks

[2], [3], [4] is an interdisciplinary approach called network

science [5]. Recent collaborative studies in network sci-

ence and neuroscience show us that CNS have features of

complex networks - such as small world topology, highly

connected hubs and modularity [6]. In another work, it

has been shown that an initially random wiring diagram

can evolve to a functional state characterized by a small-

world topology of the most strongly connected nodes and

by self-organized critical dynamics [7]. Thus, it seems that

the neural wiring problem can be reduced to the network

formation problem in which each node has discretion in

autonomously forming its links in the network relationship

[8]. Furthermore, it is widely accepted that agent-based

modeling and simulation coordinated by self-organization

and emergence mechanisms is an effective way for simulat-

ing biological systems [9] since it is possible to associate

different elements of a biological process with autonomous

computing entities called agents [10].

In this sense, a self-organized agent-based model that

learns the dynamics of CNS over time is designed and

developed. The model uses temporal data collected from

human subjects in order to estimate unknown connections

and aims to generate what is observed in natural situations.

The model has a number of properties for simulating real

biological neural networks. First, dynamic activity and spik-

ing are modeled in the individual neuron (cell) scale. This

scale was chosen since it represents the best compromise

between dynamics, complexity and observability for simulat-

ing the functional connectivity of neural networks [11]. The

dynamics of individual neurons are modeled as autonomous

agent behaviors. Driven by these autonomous behaviors,

an artificial neural network emerges through modification,

recruitment and dismissal of neurons. Secondly, the effect

of a spike on a target neuron is defined as a temporal

membrane potential change in response to the influence

of a source neuron that connects to it. That influence is

not instantaneous, and is delayed by the physical distance

between neurons (the speed of transmission is assumed

the same for all connections). Finally, to more realistically

simulate experimental conditions the model is initialized

with known connectivities, dynamic parameters and noise.

Remaining of this paper is organized as follows. The

next section gives a background information about CNS and

explains the exploration problem of synaptic functional con-

nectivity. In Section 3, the agent-based model is introduced

and explained in detail. Section 4 gives an experimental

2012 IEEE Sixth International Conference on Self-Adaptive and Self-Organizing Systems

978-0-7695-4851-7/12 $26.00 © 2012 IEEE

DOI 10.1109/SASO.2012.18

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frame and shows its results. The discussion of the results

and the approach are given in Section 6. Related work is

given in Section 7 and finally, Section 8 concludes the paper.

II. SYNAPTIC FUNCTIONAL CONNECTIVITY PROBLEM

A. The Central Nervous System (CNS)

The nervous system is a network of neurons that com-

municate information about an organism’s surroundings and

itself. A neuron is an excitable cell that processes and

transmits information by electrochemical signaling. A typ-

ical neuron can be divided into three functionally distinct

parts, dendrites, soma and axon. Roughly, the dendrites

play the role of the input device that collects synaptic

potentials from other neurons and transmits them to the

soma. The soma is the central processing unit that performs

an important non-linear processing step (integrate & firemodel): If the total synaptic potential exceeds a certain

threshold in a certain time interval (temporal integration)

and makes the neuron membrane potential to depolarize

to the threshold, then a spike with 0.5 ms duration and

100 mV amplitude is generated [12]. After emitting the

spike, the neuron membrane potential is reset to a lower

value (after hyper-polarization - AHP) from where it starts

to move towards the threshold again if there is sufficient

input current. Meanwhile, the spike is taken over by the

output device, the axon, which delivers the spike to other

neurons through synapses. A synapse is a junction between

two neurons. Most synapses occur between an axon terminal

of one (presynaptic) neuron and a dendrite of or the soma of

a second (postsynaptic) neuron, or between an axon terminal

and a second axon terminal. When a spike transmitted by

the presynaptic neuron reaches a synapse, a postsynaptic

potential (PSP) occurs on the postsynaptic neuron for 4.0 ms

(PSP duration). This PSP can either increase (excitatory PSP

- EPSP) or decrease (inhibitory PSP - IPSP) a postsynaptic

neuron’s ability to generate a spike. The PSP for a unitary

synapse can range from 0.07 mV to 0.60 mV, but normally

it is between 0.10 and 0.20 mV (see Figure 4 in [13]). The

time for a spike to reach the postsynaptic neuron involves the

axonal delay and the synaptic processing time. The axonaldelay accounts for the forward-propagation of the spike to

the synapse through the axon, while the synaptic processingtime accounts for the conduction of the spike along the

dendritic tree toward to soma.

CNS pathways basically utilize three distinct neuron

types: sensory neurons, interneurons and motoneurons. Sen-

sory neurons are responsible for converting external stimuli

from the environment into internal stimuli. Unlike other

neurons, whose inputs come from other neurons, sensory

neurons are activated by physical modalities such as light,

sound, and temperature. Interneurons form a connection

between other neurons and take part in the long loop

of the reflex events to maintain the postural balance of

the subject. Interneurons are most of the time ready to

Figure 1. Tonic firing of a neuron (modified from [14]). During tonic firing,a neuron receives continuous current and hence its membrane potential con-tinuously rises to the firing threshold and makes it fire spontaneous spikes(a). The time intervals between consecutive spikes are called interspikeintervals (ISI) and the instantaneous frequency of a spike is calculated asf = 1000/ISI . While an EPSP induces a phase forward movement of thenext spike (and thus increases the instant frequency) (b), IPSP delays theoccurrence of the next spike (and thus decreases the instant frequency) (c).

react to disturbances that could raise a reflex [15], [16],

[17]. Motoneurons project their axons outside the CNS and

control muscles. Motoneurons are tonically active and are

affected by neurons connected to them. Hundreds of EPSPs

and IPSPs from sensory neurons and interneurons arrive

at different times onto a motoneuron. This busy traffic of

inputs create the ’synaptic noise’ on the membrane of the

motoneuron. As the consequence of this noise, spikes occur

at nearly random times (Figure 1). In several intracellular

studies of tonically active motoneurons (e.g., [18], [19]), it

has been reported that the amplitude of AHP is 10 mV.

B. Exploration of Synaptic Functional Connectivity

Wiring of neurons is a key prerequisite to understand the

operation of the nervous system. While the anatomy of these

connections can be obtained by histochemical methods,

their functional connections can only be determined using

electrophysiological recordings. The functional connection

of selected sensory neurons or corticospinal fibers to mo-

toneurons can be studied directly in animal preparations.

Compared with the studies on human subjects, experiments

on animals have advantages as precise stimulation of se-

lected nerve fibers / neurons and intracellular recordings

from exact sites are possible. Therefore, interpretations of

the connections of the stimulated fibers and neurons are

straightforward and easy in animal experiments. However,

studies on experimental animals have also limitations. To

prepare animals for experiments, they have to be reduced

(anesthetized, decerebrated, sliced, etc.). It is well recog-

nized that such reductions have severe effects on synaptic

potentials. Furthermore, one need also to remember that ac-

tive involvement of the supra-spinal pathways to discharges

of motoneurons is either completely suppressed or altered

significantly in these reduced animal preparations. There-

fore, although experiments on animals deliver the influence

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of a stimulated fiber or neuron to another fiber or neuron

directly, one needs to accept the results of such experiments

with the experimental settings in mind. In particular, describ-

ing findings on reduced animal preparations as ‘functional

connections of neurons’ should be done with reservations.

Although such direct experiments cannot be performed

in humans, any experimental findings on conscious human

subjects are likely to be more functional since they are not

influenced by anesthetics or any other reduction processes

and since the activity of supra-spinal centers are maintained.

To study the functional connection of neurons in human sub-

jects it has been customary to use stimulus-evoked changes

in the discharge probability and rate of one or more motorunits in response to stimulation of a set of sensory neurons

(peripheral afferents) or corticospinal fibers. These are the

most common ways to investigate the workings of peripheral

and central pathways in human subjects. Although these are

indirect methods of studying the human nervous system, they

are nevertheless extremely useful as there is no other method

available yet to record synaptic properties directly in human

subjects.

Motor units are composed of one or more motoneurons

and the corresponding muscle fibers they innervate. When

motor units are activated, all of the muscle fibers they

innervate contract. Likewise, stimulation of a sensory-motor

peripheral nerve fiber produces similar contractions. Basi-

cally, a sensory-motor peripheral nerve fiber is composed of

the following components (from largest to narrowest): group

Ia and group Ib sensory axons, alpha motor axons; group II,

group III and group IV sensory axons and gamma motor

axons. A nerve fiber’s threshold to electrical stimulation is

inversely proportional to the diameter of its axon, larger

axons are more sensitive to electrical stimulation. In this

sense, while in a low-threshold stimulation experiment only

group Ia and group Ib sensory axons are activated, in a

high-threshold stimulation experiment all the sensory axons

can be activated. Additionally, it is known that group Ia

afferents make monosynaptic connections, group Ib sensory

neurons make disynaptic connections and group II sensory

neurons make polysynaptic connections with the alpha mo-

tor neurons. The output from the motor units is through

the motoneurons, and is measured by reflex recordings

from the muscle. However, most of the synaptic inputs to

motoneurons from sensory neurons do not go directly to

motoneurons, but rather to interneurons that synapse with

the motoneurons. And the synaptic connectivity of these

interneurons is still not known totally.

C. Functional Connectivity Analysis of Motor Units

The ability to record motor unit activity in human subjects

has provided a wealth of information about the neural control

of motoneurons, and in particular has allowed the study of

how reflex and descending control of motoneurons change as

a function of task, during fatigue and following nervous sys-

tem injury. Although synaptic potentials cannot be directly

recorded in human motoneurons, their characteristics can

be inferred from measurements of the effects of activating

a set of peripheral or descending fibers on the discharge

probability of one or more motoneurons. Recently, these

effects are being assessed by compiling a frequency-gram

(PSF) that uses the instantaneous discharge rates of single

motor units for estimating the synaptic potentials produced

by afferent stimulation by neuroscientists (for review see

[20]). The PSF plots the instantaneous discharge rate values

against the time of the stimulus and is used to examine

reflex effects on motoneurons, as well as the sign of the net

common input that underlies the synchronous discharge of

human motor units (Figure 2a). The instantaneous frequency

values comprising the PSF should not necessarily be affected

by previous (prestimulus) activity at any particular time.

However, since the discharge frequency of a motoneuron

reflects the net current reaching the soma [21], any signifi-

cant change in the poststimulus discharge frequency should

indicate the sign and the profile of the net input.

To determine significant deflections, the cumulative sum

(CUSUM) of PSF record is used (Figure 2b). The CUSUM

is calculated by substracting the mean pre-stimulus baseline

from the values in each bin and integrating the remainder

[22]; PSP-induced effects are considered significant if the

post-stimulus CUSUM values exceed the maximum pre-

stimulus CUSUM deviation from zero (i.e. the error box

[23], [24], indicated by the horizontal lines in Figure 2b).

As can be seen from this figure, there is an early and long-

lasting excitation (LLE) indicated by the increased frequency

from about 40 ms poststimulus to about 100 ms. After this

LLE there is a period of long-lasting inhibition (LLI) going

from about 100 ms poststimulus to about 300 ms. However,

since after 200 ms of stimulation the subject is able to

change the discharge rate of his/her motor unit, the events

later than 200 ms cannot be considered as reflex events. Only

before 200 ms of poststimulus discharge rates might give an

exact information about the network of the motor unit.

III. SIMULATING WITH SELF-ORGANIZING AGENTS

To tackle with the synaptic connectivity problem, we

have selected the agent-based simulation of the system [8]

with self-organizing agents using the Adaptive Multi-Agents

Systems (AMAS) approach [25].

A. The AMAS Approach

In the AMAS approach, the system is composed of a

set of dynamic number of agents A = {a0, a1, ...}. In this

approach, a system is said to be functionally adequate if it

produces the function for which it was designed, according

to the viewpoint of an external observer who knows its

finality. To reach this functional adequacy, it has been proven

that each autonomous agent ai ∈ A must keep relations as

cooperative as possible with its social (other agents) and

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Figure 2. PSF and PSF-CUSUM from motoneuron responses to single PSP recorded from human soleus muscle.

physical environment [25], [26]. The configuration of inputs

coming from other agents (and physical environment) leads

ai to produce a new decision. A non-desired configuration

of inputs causes a non-cooperative situation (NCS) to occur.

ai is able to memorize, forget and spontaneously send

feedbacks related to desired or non-desired configurations

of inputs coming from other agents. We denote the set of

feedbacks as F and model sending a feedback fa ∈ F using

the action of the form send(fa,R) where a is the source of fand receiver agents R ⊂ A \ {a}. A feedback fa ∈ F can

be about increasing the value of the input (fa↑), decreasing

the value of the input (fa↓) or informing that the input is

good (fa≈).

When a feedback about a NCS is received by an agent,

at any time during its lifecycle, it acts in order to avoid or

overcome this situation [27] for coming back to a coopera-

tive state. This provides an agent with learning capabilities

and makes it constantly adapt to new situations that are

judged harmful. In case a NCS cannot be overcome by an

agent, it keeps track of this situation by using a level of

annoyance value ψfa where fa is the feedback about this

NCS. When a NCS is overcome, ψfa is set to 0, otherwise

it is increased by 1. The first behavior an agent tries to

adopt to overcome a NCS is a tuning behavior in which

it tries to adjust its internal parameters. If this tuning is

impossible because a limit is reached or the agent knows

that a worst situation will occur if it adjusts in a given way,

it may propagate the feedback (or an interpretation of it)

to other agents that may handle it. If such a behavior of

tuning fails many times and ψfa crosses the reorganization

annoyance threshold ψreorganization (reorganization condi-

tion), an agent adopts a reorganisation behavior in which

it tries to change the way in which it interacts with others

(e.g., by changing a link with another agent, by creating a

new one, by changing the way in which it communicates

with another one and so on). In the same way, for many

reasons, this behavior may fail counteracting the NCS and a

last kind of behavior may be adopted by the agent: evolutionbehavior. This is detected when ψfa crosses the evolution

annoyance threshold ψevolution (evolution condition). In the

evolution step, an agent may create a new one (e.g., for

helping itself because it found nobody else) or may accept to

disappear (e.g., it was totally useless and decides to leave the

system). In these two last levels, propagation of a problem

to other agents is always possible if a local processing is

not achieved. The overall algorithm for suppressing a NCS

by an agent is given in Algorithm 1.

Algorithm 1 Cooperative behaviors of an agent ai upon

receiving a feedback faj about a NCS of the agent aj .

1: if ψfaj< ψreorganization // tuning condition

2: if tuning behavior succeeds ψfaj← 0

3: else ψfaj← ψfaj

+ 1 and send(fj , R) endif4: else if ψfaj

< ψevolution // reorganization condition

5: if reorganization behavior succeeds ψfaj← 0

6: else ψfaj← ψfaj

+ 1 and send(faj , R) endif7: else // evolution condition

8: if evolution behavior succeeds ψfaj← 0

9: else send(faj , R) endif10: endif

The AMAS approach is a proscriptive one because each

agent must first of all anticipate, avoid and repair a NCS.

Thus, the designer, according to the problem to be solved,

has: 1) to determine what an agent is, then 2) to define

the nominal behavior which represents an agent’s behavior

when no NCS exist, then 3) to deduce the NCSs the agent

can be faced to, and finally 4) to define the cooperative

behavior (tuning, reorganization and/or evolution) the agent

has to perform when faced to each NCS in order to come

back to a cooperative state. This is the process adopted in

the following of this section. Moreover, to build a real self-

adaptive system, the designer has to keep in mind that agents

only have a local view of their environment and that they do

not have to base their reasoning on the collective function

that the system must achieve.

B. Identification of agents and their nominal behaviors

We model the agent-based simulation model Sim ba-

sically capturing all taken design decisions based on the

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AMAS theory as Sim = (G, ν) where G is the agent-based

neural network and ν is the external observer agent of G.

1) Agent-based Neural Network Model: We model the

neural network as a dynamic directed graph G(t) =(N (t),S(t)) where N (t) ⊂ A denotes the time varying

neuron agent (vertex) set and S(t) denotes the time vary-

ing synapse (edge) set. The set of excitatory (respectively

inhibitory) neuron agents at time t is denoted by N+(t)(resp. N−(t)) where N (t) = N+(t)∪N−(t). One nominal

behavior of neuron agents is spike firing. A neuron agent

n fires a spike when its membrane potential pn crosses

the firing threshold. We define N spike(t) to be the set of

neuron agents that fired their last spike at time t. We also

denote tn for indicating the last spike firing time of the

neuron agent n where n ∈ N spike(tn). When a neuron agent

n fires a spike, this spike is emitted through its synapses

to the postsynaptic neuron agents. We denote the set of

presynaptic neighbors of a neuron agent n at time t as

Pren(t) = {m ∈ N (t)|{m,n} ∈ S(t)} and the set of

postsynaptic neighbors of a neuron agent n ∈ N at time tas Postn(t) = {k ∈ N (t)|{n, k} ∈ S(t)}. Apart from pre-

and postsynaptic neighbors, a neuron agent has also another

type of neighborhood which contains all the neuron agents

it has contacted during its lifetime, even the synapses in

between removed after some time. Formally, a friend agent

m of a neuron agent n at time t is denoted by there exists

some time t′ ≤ t such that m ∈ Pren(t′) or m ∈ Postn(t′).All friends of n at time t is denoted by Friendn(t).The set of presynaptic neuron agents that contributes to

the activation of a postsynaptic neuron n at time tn is

modeled as Contn(tn) where Contn(tn) ⊂ Pren(tn) and

tn > tk > tn − 4.0 for all k ∈ Contn(tn). Lastly, a neuron

agent has to know the friends who activated temporally

closest to its activation. Formally, a temporally closest friend

agent m of a neuron agent n at time t is denoted by there

exists some time t′ < tn ≤ t such that m ∈ Friendn(t),t′ = tm and there is no t′ < t′′ < tn such that for all

k ∈ Friendn(t), t′′ = tk. All temporally closest friends of

n at time t is denoted by Tempn(t).A synapse {n,m} conducts a spike from n to m through

the interval [tn, t′] if n ∈ N spike(tn), and t′ = tn + dnm

where dnm is the delay for delivering the spike from n to

m. We denote the spike delay as dnm = daxnm + ddennm where

daxnm is the axonal delay of {n,m} and ddennm is the synapticprocessing time. We assume that ddennm = 0.5 ms [28]. daxnm,

on the other hand, may change depending on the length

and type of the axon. We also say that a synapse {n,m}potentiates (respectively depresses) the membrane potential

p of m with a synaptic strength η at time t′ where 0.07 ≤|η| ≤ 0.60 during the PSP duration dpsp = 4.0 if n ∈ N+

(resp. n ∈ N−) and m is not removed at any time during

the interval [t′, t′ + dpsp].We model the set of sensory neuron agents at time t as

K(t) ⊂ N+(t) where for all n ∈ K(t), we have Pren(t) =

∅ and Postn(t) = ∅. Since Pren(t) = ∅, they have a

nominal action of the form activate() triggered by the viewer

agent (see next subsection) in order to be able to fire.We model the set of motoneuron agents at time t as

M(t) ⊂ N+(t) where for all n ∈ M(t), we have

Pren(t) = ∅ and Postn(t) = ∅. In the current model there

is only one n ∈ M(t) since the focus is on single motor

units. Apart from the integrate & fire nominal behavior, a

motoneuron agent n continuously increases its membrane

potential pn with Δp in order to imitate tonic firing behavior.Finally, we model the set of interneuron agents as I ⊂ N

where for all n ∈ I , we have Pren = ∅, Postn = ∅ and

dax = 0 since their axonal delays are extremely low.2) The viewer agent: The viewer agent ν is designed

to trigger the recruitment of synaptic connections and the

functional connectivity of the agent-based neural network

G. It acts like a surface electrode and gives inputs to Gby coordinating random activation of all sensory neuron

agents s ∈ K. Meanwhile, it monitors and records the

outputs of the motoneuron agent m that take place over

time to compare them with reference data. This comparison

takes place between the latency of the beginning (lbegin)

and the end (lend) of the network. These parameters are

given to ν initially, and the other outputs of m are regarded

as its unaffected behavior. When an output is observed by

ν at time t, first the latency of this output (lcurrent) is

calculated (Algorithm 2, line 1), then the observed output

is compared to the reference output observed at the same

latency (Algorithm 2, line 4) with a tolerance of τ , and

finally an appropriate feedback is sent to m (Algorithm 2,

lines 5, 6 and 7). According to this comparison, ν makes

assessments about the behavior of m for detecting if it is

functionally adequate to the real motoneuron or not. If the

observed output is generated between lbegin and lend where

lend > lbegin > 0, ν sends a feedback f ∈ Fif to m.

Otherwise, ν does not send any feedback to m.

Algorithm 2 Response of viewer agent ν upon monitoring

an output at time t from motoneuron m where tsti is the

last stimulation time of sensory neurons.

1: lcurrent ← (t - tsti)2: if lbegin ≤ lcurrent ≤ lend3: generate the simulated PSF

4: calculate the difference ε of PSFs at time lcurrent5: if ε > τ then send(fν ↓,m)6: else if ε < τ then send(fν ↑,m)7: else send(fν ≈,m) endif8: endif

Additionally, ν is responsible for stopping the simulation

run when the evolution of the neural network ends. ν detects

this situation by evaluating the output of m. Formally, G is

said to be stable at time t′ if for all t1, t2 ∈ R+ where

t2 > t1 ≥ t′, for all feedbacks f , we have f ∈ Fif ≈.

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C. Identification of Non-Cooperative Situations

The proposed agent-based neural network model, in which

neuron agents and synapses can be inserted or removed, is

subjected to NCSs. All NCSs are identified by analyzing the

possible bad situations of real human single motor units.

1) Bad Temporal Integration: The temporal integration

of the inputs provided by synapses affects what a neuron

agent does. For an interneuron agent n, these inputs affect

whether n can fire or not, while for a motoneuron agent

m they affect the frequency of the spikes of m. Sensory

neurons, however, never detect this situation since they do

not need input neurons in order to fire. Consequently, when

this temporal integration is bad, a neuron agent either cannot

fire or have a bad firing behavior. When such a situation

is detected at time t, the neuron agent should improve its

existing inputs or should search for new inputs with the right

timing. To do so, it sends a temporal integration increaseor decrease feedback (f ∈ Fti↑ or f ∈ Fti↓) to some or

all of its neighbor neuron agents. Otherwise, the temporal

integration is good and a temporal integration good feedback

(f ∈ Fti≈) is sent to Pren(t).

An interneuron agent detects a bad temporal integrationNCS at time t if during the interval [t−Δtmax, t] it did not

generate any spikes where Δtmax is the maximum time slice

an interneuron agent can stay without spiking. However, the

motoneuron agent m is unable to detect the same situation

by itself. It detects when it receives an instant frequency

feedback fν ∈ Fif at time t about to its last spike at time

tm (see Section III-C2). Since this spike is related to the

temporal integration of its own membrane potential and its

presynapses, there are two cases: (1) Contm(tm) = ∅, and

(2) Contm(tm) = ∅. In the first case, m sends a feedback

fm ∈ Fif to its temporally closest friend neurons to be

able to have contributor synapses. In the second case, the

problem can be turned into a temporal integration problem

and a temporal integration feedback fm ∈ Fti is sent to its

temporally closest friends Tempm(t).

2) Bad Instant Frequency: A motoneuron agent fires

continuously and its firing behavior might be affected by

its pre-synapses when a stimulation is given to the sensory

neuron agents. The motoneuron agent is expected to gen-

erate frequencies similar to the reference data. When the

motoneuron agent emits a spike, the viewer agent observes

it and calculates the instant frequency value for that spike

using the previously emitted spike. However, it is not log-

ical to compare an individual frequency information to the

reference data since there can be many frequency values

at a specific time and the reference data contain the noisy

behavior of the motoneuron. In this sense, to reduce the

noise and to facilitate the comparison, the moving average

frequency values are used. Consequently, the average fre-

quency at time of spike is expected to be close enough to

the average frequency of the reference data. As a result of

this comparison, the viewer sends an instant frequency isgood, increase or decrease feedback (fν ∈ Fif↑, fν ∈ Fif↓or fν ∈ Fif≈) to the motoneuron agent.

D. Cooperative Behaviors

The tuning behavior of neuron agents is modelled using

the action of the form tune({n,m}, f ) for n,m ∈ N (t) and

f ∈ F , which correspond to the adjustment of {n,m}.η by fat time t. An autonomous and cooperative neuron agent must

be able to decide by itself the modification of its synapse.

Thus, this action can only be executed by n over {n,m}.Moreover it is assumed that no opposite adjustment is done

at the same time. The reorganization behaviors of neuron

agents are modeled using actions of the form add({n,m})and remove({n,m}) for n,m ∈ N (t), which correspond to

the formation and suppression (respectively) of {n,m} at

time t. It is assumed that no synapse is both added and

removed at the same time. The evolution behaviors of neuron

agents are modelled using actions of the form create(n,m),createInverse(n,m) and remove(n) for n,m ∈ N , which

correspond to the creation and suppression (respectively) of

neuron agents. It is assumed that no neuron agent is both

added and removed at the same time.

NCSs are suppressed by processing the aforementioned

actions as described in the following subsections.

Algorithm 3 Response to the feedback fm received at time

t of neuron agent n where fm ∈ Fif and m ∈ M(t) and

n ∈ N (t).

1: � evolution condition

2: if n ∈ N+ then create(Pren ∪ n,m) endif

1) Suppression of “Bad Instant Frequency” NCS: When

the motoneuron agent m receives fν ∈ Fif from ν about

its last spike at time tm, it evaluates fν taking into account

Prem(t) in a temporal manner. There might be two cases

depending upon some n ∈ Prem(t) that affects its last spike

in the right time exists or not. If there exists some n ∈Prespikem (t′) where t′ = tm − 0.5, m turns the problem

into a temporal integration problem and sends fm ∈ Fti

to all n ∈ Prem(t′). Otherwise, m sends fm ∈ Fif to

Prespikem (t′) where Prespikem (t′′) = ∅ for all tm ≥ t′′ > t′.When a feedback fm ∈ Fif is received by a neuron agent n,

since it cannot help m by tuning or reorganization, it directly

executes its evolution behavior (Algorithm 3): it creates an

excitatory interneuron agent k by including itself as one of

the presynaptic neurons of k. Therefore k will likely fire

after n and there will be a time shift in the network.2) Suppression of “Bad Temporal Integration” NCS:

When a feedback fm ∈ Fti is received by a neuron agent

n, it first tries to tune the synapse {n,m} if m ∈ Postm(Algorithm 4, line 2). If n cannot help m by tuning, it tries

reorganization: either by adding a synapse in between if

there is no synapse, or removing the existing synapse. A

16

Algorithm 4 Response to the feedback fm received at

time t of neuron n where fm ∈ Fif and m,n ∈ N (t).When n is unable to help, it propagates the feedback

fm to all its temporally closest friends Tempn(tn) using

send(fm, T empn(tn)).1: � tuning condition

2: if m ∈ Postn then tune({n,m}, fm) endif3: � reorganization condition

4: if m ∈ Postn then5: if ((n ∈ N+ ∧ fm ↓ ) ∨ (n ∈ N− ∧ fm ↑ )) then6: remove({n,m}) endif7: else // m /∈ Postn8: if ((n ∈ N+ ∧ fm ↑ ) ∨ (n ∈ N− ∧ fm ↓ )) then9: add({n,m}) endif

10: � evolution condition

11: if m ∈ Postn then12: if ((n ∈ N+ ∧ fm ↑ ) ∨ (n ∈ N− ∧ fm ↓ )) then13: create(Pren,m) endif14: else // m /∈ Postn15: if (n ∈ N+ ∧ fm ↓ ) ∨ (n ∈ N− ∧ fm ↑ ) then16: createInverse(Pren,m) endif

neuron n ∈ N+ (respectively n ∈ N−) adds a new synapse

(Algorithm 4, line 10) if fm ↑ (resp. fm ↓ ) and removes

the existing synapse (Algorithm 4, line 6) if fm ↓ (resp.

fm ↑ ). As a last resort, n creates a new neuron for helping

m. A neuron n ∈ N+ (respectively n ∈ N−) creates a new

neuron n ∈ N+ (resp. n ∈ N−) (Algorithm 4, line 15) if

fm ↑ (resp. fm ↓ ) and creates a new neuron n ∈ N− (resp.

n ∈ N+) (Algorithm 4, line 19) if fm ↓ (resp. fm ↑ ).

As mentioned before, the bad temporal integration NCS

can be detected by both interneuron agents and the mo-

toneuron agent. When detected by an interneuron agent,

any synaptic strength change within a 4 ms time range is

welcome, since the objective is to activate the interneuron

agent. Thus, when a neuron agent cannot suppress such a

situation, it asks its presynaptic or postsynaptic neighbors.

For the motoneuron agent, however, the objective is to have

a synaptic strength change at a specific time. Thus, when

a neuron agent cannot suppress such a situation, it asks its

temporally closest friend neurons. This way, the feedback

propagates through the network.

IV. EXPERIMENTAL FRAME

The proposed model was implemented using RePast Sym-

phony 2.0.0 beta, an agent-based simulation environment

written in Java [29]. This model was then verified and

validated by using the model testing framework given in

[30]. The dynamic parameter of the synapses, the strength,

is implemented as described in [31]. The model proceeds

0.5 ms time steps (tick). The suitable parameter space was

determined in preliminary investigations.

A. AMAS Parameters

The reorganization annoyance threshold is set to 20

(ψreorganization = 20) and the evolution annoyance thresh-

old is set to 40 (ψevolution = 40).

B. Initial Scenario

To test our model we have chosen to simulate the neural

circuitry of single motor units using the data obtained from

low-threshold stimulation experiments on human soleus

muscles (Figure 2a). The only a priori information about sin-

gle motor units is that Ia sensory neurons make monosynap-

tic connections with the motoneuron (see Section II-B). In

this sense, we considered this path as the shortest path in our

network and defined its duration as l. Thus, we initialized the

simulations as N (0) = {s,m}, S(0) = {{s,m}, {m,∅}}and d{s,m} = d{m,∅} = l/2 where s ∈ K, m ∈M and l is

the latency of the beginning of LLE extracted from the PSF-

CUSUM of the reference experimental data (Figure 2b). In

this sense, lbegin is set to l since the earliest stimulus-evoked

change of the motoneuron behavior can be observed at l and

lend is set to 200 (see Section II-C).

C. Data Configuration

All the data used in these experiments were obtained

by recording the single motor unit activity from human

subjects. To be able to determine the similarity between

the reference and the simulated data, both of them were

converted into their respective moving average frequencies.

The frequency of firing of motor units integrates the exci-

tatory and inhibitory synaptic activities [21] and involves

the noisy behavior of the motoneuron. The moving aver-

age frequencies reduce the noise of the motoneuron and

represent the net effect on the respective motoneuron. Any

significant increase1 (respectively decrease) in the average

frequency of the motor unit represents a stimulus-induced

net excitatory (resp. inhibitory) effect. Therefore, each time

the motoneuron agent emits a spike, the average frequency

for that spike is calculated and compared to the reference

average frequency and a proper feedback is sent to the

motoneuron agent by the viewer agent.

In order to provide a good feedback mechanism, the

moving average frequency diagram was developed in 3

stages: (1) the bin size is set to 0.5 and the bins in the PSF

diagram with no frequency value were given the values in the

preceding bin. This was to ensure that the average frequency

value did not suddenly drop down to 0 Hz in these bins

[32]. This approach assumed that empty bins represented

the same frequency as the preceding bin even though they

failed to be filled due to the low number of trials and/or

due to the chance. (2) Then the raw record of the PSF was

modified by averaging the frequency values in each bin (600

1A significant change in the activity is a change which is significantcompared to the prestimulus activity.

17

Table IRESULTS FOR 10 SIMULATION EXPERIMENTS

Similarity (r2) Neuron agents Synapses Convergence in ticks

0.9093 621 1013 97,201,624

0.9159 778 1307 202,681,101

0.9179 611 993 100,400,005

0.9186 680 1121 123,279,523

0.9214 564 905 97,003,656

0.9348 583 935 94,000,891

0.9351 769 1292 158,718,110

0.9521 664 1085 122,630,387

0.9586 553 875 81,120,244

0.9592 611 992 116,481,384

ms bandwidth). (3) The averaged frequency record was then

filtered by the moving averager using 4 bin average and

smoothed 8 times.

D. Tonic Firing Configuration

For imitating the nominal behavior of the biological mo-

toneuron, the motoneuron agent m uses the reference data.

When the reference data are provided to m, it removes the

discharge rate values after stimuli and calculates a statistical

distribution using the remaining part. Then using the statis-

tical parameters of the distribution, a frequency generator,

which is used to generate consecutive interspike interval

(ISI) values for m, is created. Each time a new ISI value

is calculated using this generator, the instant membrane

potential increase Δp is calculated as Δp = AHPm/ISIand at each tick, pm is increased by Δp ∗ tick. For the

statistical calculations, the SSJ2 (Stochastic Simulation in

Java) library is used. To increase the reliability of the

motoneuron agent, the goodness-of-fit test for the tonic firing

behavior is also performed using the aforementioned testing

framework [30]3.

V. SIMULATION RESULTS

After the simulation ends, the results are analyzed in

order to ensure that the generated network is functionally

equivalent to the reference real network. To calculate the

similarity of two networks, a cross-correlation analysis is

performed between the simulated PSF-CUSUM and the

reference PSF-CUSUM. The correlation r will yield a 0

when there is no correlation (totally uncorrelated) and a

1 for total correlation (totally correlated). The degree of

similarity is then calculated as r2. This information is fair

enough in order to claim that the two underlying networks

are functionally equivalent or not.

In 10 simulation experiments (Table I), the functional

similarity observed is between 90.93% and 95.92% (average

2http://www.iro.umontreal.ca/˜simardr/ssj/indexe.html, last access on 14July 2012.

3The case study given in [30] is comprehensively describing this test.

93.48%), the number of neuron agents is between 553 and

778 (average 643.4) and the number of synapses is between

875 and 1307 (average 1051.8). Figure 3 shows the PSF and

the PSF-CUSUM diagrams of a simulation run.

VI. DISCUSSION

The results are very promising. They show that the

developed model is able to learn and simulate the functional

behavior of human single motor units. The average number

of neuron agents observed at the end of simulations is not

an odd value for a single motor unit pathway. However, it is

unclear that this is the case. Moreover, the model does not

simulate the functional behavior smoothly (see Figure 3b).

This is probably caused by the lack of proper feedbacks

(see the gaps between discharge rates in Figure 3a), since

the viewer agent is sending feedbacks in response to the

observed outputs. Besides, according to the AMAS theory,

giving right feedbacks to the right agents is important. Thus,

the calculation of the average PSF must be done carefully

since it affects right recruitment of the network. It should

be done slightly without displacing the peaks and throughs

significantly. During our preliminary investigations, it has

been observed that when the average PSF is smoothed

more, the information about the dynamics of the system

is lost, and as long as it is smoothed less, the noise of

the motoneuron agent prevents the interneuron agents to

learn the right dynamics. The tolerance value τ is also

an important parameter closely related to right feedbacks.

Similar to smoothing, when τ is higher, the information

about the dynamics of the system is lost, and when τ is

smaller, it is harder to detect good outputs.

Another important aspect is the convergence efficiency of

the model. Parameters that affect the convergence are the

annoyance thresholds and the AHP level of interneurons.

The annoyance thresholds are set to high values to increase

the propagation of the feedbacks inside the network. When

these parameters are lower, the agents tend to make reorga-

nization and evolution more quickly and the perturbation of

the system increases. Notwithstanding, when they are higher,

although the dynamics of the network does not change

and the network successfully converges, the convergence

efficiency decreases. For computational simplicity and for

increasing the convergence speed, we allowed the AHP level

of interneurons a low value, so that they can activate with

at least two synapses. However, it is unclear that this should

be the case for real networks.

VII. RELATED WORK

Buibas et al. [11] present a formal modeling framework

for using real-world data to map the functional topology

of complex dynamic networks. The framework formally

defines key features of cellular neural network signalling

and experimental constraints associated with observation

and stimulus control, and can accommodate any appropriate

18

Figure 3. The simulated PSF (a) and PSF-CUSUM (b) compiled from the motoneuron agent responses of a simulation experiment. The functionalsimilarity between the simulated network (b) and the real network (Figure 2b) is 95.92 %.

model intracellular dynamics. They claim that, the frame-

work is particularly well-suited for estimating the functional

connectivity in biological neural networks from experimen-

tally observable temporal data. However, the framework

is unable to estimate and map the functional topology of

complex networks with unknown connectivities. There are

also approaches for estimating parameters and dynamics of

small groups of neurons [33], [34], Eldawlatly et al. [33]

use dynamic bayesian networks in reconstructing functional

neuronal networks from spike train ensembles. Their model

can discriminate between mono- and polysynaptic links

between spiking cortical neurons. Makarov et al. [34] present

a deterministic model for neural networks whose dynamic

behavior fits experimental data. They also use spike trains

and their model permits to infer properties of the ensemble

that cannot be directly obtained from the observed spike

trains. However, these studies are far from establishing a

complete functional connectivity of larger networks such as

single motor unit pathways.

There are also simulators focused on creating a wiring

diagram of all the neurons in the brain [35], [36]. The

Human Connectome Project (HCP) [35] is an ambitious

effort to map the neural pathways that underlie human brain

function. The HCP proposes to resolve this by using new-

generation magnetic resonance imaging (MRI) machines,

like that used to scan my brain, to trace the connectomes

of more than 1,000 individuals. The Human Brain Project

[36] aims to create a wiring diagram of all the neurons

in the brain, and neuroscientists have developed innovative

techniques for automatically imaging slices of mouse and

cat brain, yielding terabytes of data so far. However, it

is not proven that MRI techniques can produce a reliable

picture of normal connectivity, never mind the types of

abnormal connection likely to be found in brain disorders,

and some researchers argue that the techniques have not been

adequately validated [37].

VIII. CONCLUSIONS

Although there are studies aimed at simulating biological

neural networks in the literatureIn this study we have, for

the first time, generated artificial neural networks for human

single motor unit pathways using PSF data through a self-

organization process. The generated networks are modified

through NCSs of agents. Driven by intermittent activations

of sensory neuron agents and the spontaneous activity of the

motoneuron agent, an artificial neuronal pathway emerges

through recruitment, dismission and modification of neuron

agents and synapses. Our present findings do not constitute

a proof that the simulated neural network is exactly as the

real one. However, it is shown that the developed simulator is

able to generate what is observed in natural situations. Since

the nominal behaviors of the model elements conform to the

biological elements (they are all verified and validated), it

can also be said that the simulated network is biologically

plausable. This appears as an arresting conclusion that makes

our understanding about synaptic functional connectivity

of human motor units more clear. Nevertheless, a detailed

analysis of the generated network should be done in order to

prove this conclusion. As a result of this analysis the number

of neuron agents should be optimised if necessary.

To increase reliability of the model, we are planning to

use single motor unit data recorded from other muscles (e.g.

human tbialis anterior muscle). Furthermore, it would be

interesting to continue research along multiple motor unit

pathways, to see to what extend the pathways in humans

can be simulated.

ACKNOWLEDGMENT

Onder Gurcan is supported by the Turkish Scientific

and Technological Research Council (TUBITAK) through

a domestic PhD scholarship program (BAYG-2211) and

the French Government through the cotutelle scholarship

program.

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