Microsoft Word - behavior1.doc
Behavioral response to strong aversive stimuli: A neurodynamical 
model 
Kaushik Majumdar, Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai – 600113, India. 
E-mail: kaushik@imsc.res.in  
Abstract 
In this paper a theoretical model of functioning of a neural circuit during a behavioral response has been 
proposed. A neural circuit can be thought of as a directed multigraph whose each vertex is a neuron and 
each edge is a synapse. It has been assumed in this paper that the behavior of such circuits is manifested 
through the collective behavior of neurons belonging to that circuit. Behavioral information of each neuron 
is contained in the coefficients of the fast Fourier transform (FFT) over the output spike train. Those 
coefficients form a vector in a multidimensional vector space. Behavioral dynamics of a neuronal network 
in response to strong aversive stimuli has been studied in a vector space in which a suitable pseudometric 
has been defined. The neurodynamical model of network behavior has been formulated in terms of existing 
memory, synaptic plasticity and feelings. The model has an analogy in classical electrostatics, by which the 
notion of force and potential energy has been introduced. Since the model takes input from each neuron in a 
network and produces a behavior as the output, it would be extremely difficult or may even be impossible 
to implement. But with the help of the model a possible explanation for an hitherto unexplained 
neurological observation in human brain has been offered. The model is compatible with a recent model of 
sequential behavioral dynamics. The model is based on electrophysiology, but its relevance to 
hemodynamics has been outlined. 
Keywords: Theoretical neuroscience, neural circuit, spike train, FFT, pseudometric space, dynamical 
system, decision making, electrophysiology, hemodynamics. 
1. Introduction 
     Quick behavioral response to strong aversive stimuli (such as threat from a predator or 
an imminent danger of being hurt) is a key to survival throughout the animal kingdom. 
Network models of animal behavior have been elaborately discussed in (Schmajuk 1997). 
A neural network model of rats’ anxiety behavior had been studied by Salum and 
colleagues (Salum et al. 2000), but they didn’t take into account the functioning of the 
nerve cells in the brain during the task performance. Recently a neurodynamical model 
for conditional visuomotor association task has been proposed (Loh & Deco 2005). In 
this model a trial and error paradigm has been assumed in a stochastic decision space. An 
integrate and fire neuronal network model has been proposed to realize the paradigm. A 
neural network model of brain or cognitive state machine (CSM) to study decision 
making in a competitive environment has also been proposed (Rabinovich et al. 2006). 
The dynamics of making a choice from among multiple conflicting options has been 
formulated by Lotka-Volterra type of equations. It is evident that intelligent decisions in a 
sequential behavior have to be stable against noise and reproducible to allow 
memorization and reuse of successful decision sequences in the future. On the other 
hand, it also has to be sensitive to new information from the environment. These two 
fundamentally contradictory requirements have been taken care of in (Rabinovich et al. 
2006). 
     In this paper a neurodynamical model of response behavior of a neural network to 
strong aversive stimuli has been presented. The assumption is that the network will 
behave in a manner to avoid repeating the negative experiences of the past. No specific 
neuronal network model has been assumed. In this approach information has been 
extracted (at least theoretically) from a behaving neuronal network by FFT on spike 
trains of all the neurons in the network, which should work across all networks of 
neurons irrespective of their architecture. The neurodynamical system has a unique 
representation in  the information space where the Fourier coefficients of a spike train are 
arranged as a vector and uniquely represent the spike train. All the calculations have been 
carried out in that space. The model depends on past memory, synaptic plasticity and 
intensity of feeling. Interestingly, a short latency (120 – 160 ms) had been reported before 
response to aversive stimuli in the right prefrontal cortex of a human subject. No such 
latency was observed in case of pleasant or neutral stimuli (Kawasaki et al. 2001). The 
present model can offer a possible explanation for this apparently perplexing 
phenomenon. The following assumptions have been made: 
1) Structure: Any behavior involves a network in the nervous system which if 
represented as a directed multigraph will have neurons as vertices and synapses as 
(directed) edges. 
2) Function: The functional behavior of this network is manifested by the collective 
behavior of the neurons present in the network. 
3) Memory: Memory of experiences of past behaviors of this network is stored 
entirely within the network and nowhere out side of it. 
4) Plasticity: Plasticity of each synapse is a time dependent function (called synaptic 
weight) and the total plasticity of the network between any two behaviors is also a 
time dependent function called network plasticity. 
5) Feeling: Feeling is associated with each behavior as a specific mathematical 
function which controls how experience associated with this behavior will mediate 
the intensity of effect of this behavior on any other behavior. 
6) Interaction: Network plasticity mediates the interaction of the ongoing behavior 
with the memory of past behaviors stored in the network. 
In section 2 each of the above points will be explained briefly and will be represented 
with appropriate mathematical expressions and equations. In section 3 with the help of an 
analogy from classical electrostatics the behavior dynamics of the network in terms of 
those expressions and equations will be formulated. Notion of a force like and a potential 
energy like expressions have been introduced. Despite computational difficulties 
involved with the model, in section 4 it has been related to reality first by offering an 
explanation to an hitherto unexplained observation in the human brain, second relating it 
with a successful decision making model and third relating this electrophysiological 
model to hemodynamical activity of the brain. The paper will be concluded with 
discussions and future directions. 
2. Modeling of the parts 
A) Structure 
     The closest computational analog of a neuronal circuit is a directed multigraph, whose 
each node will represent a neuron and each edge a synapse (Majumdar & Kozma, 2006). 
It can be represented as a three dimensional array ],,[ kjia . If there are p  different 
synapses joining the neuron i  with the neuron j  (assuming that all neurons in the brain 
have been numbered) then ],,[ kjia  will give the weight (the gain in synaptic 
transmission) of the kth  synapse joining the neuron i  with the neuron j , where 
pk ≤≤1 . If the neuron i  and neuron j  are excitatory then ],,[ kjia  will be positive, if 
they are inhibitory then ],,[ kjia  will be negative. For given values of ji,  and k  ],,[ kji  
uniquely determines a synapse and ],,[ kjia  represents the synaptic weight. In general 
],,[ kjia  will be a function of time and may be written as )](,,[ tkjia  or )(taijk , where t  
denotes time. 
B) Function 
     Computations in the central nervous system (CNS) have been viewed from three 
different angles – (a) synaptic computation (Abbott & Regehr 2004), (b) dendritic 
computation (London & Hausser 2005) and (c) neuronal computation (Koch 1999; 
Borisyuk & Rinzel 2005). In a neuronal network synapses are activating the neurons and 
neurons are activating the synapses. In this sense the aggregate behavior of the neurons in 
a network is equivalent to aggregate behavior of the synapses in that network. On an 
average every neuron receives input from about 
10  synapses and therefore from the 
computational modeling purpose it would be more convenient to consider the aggregate 
behavior of neurons rather than the aggregate behavior of synapses in a network. The 
behavior of a neuron during a particular epoch of time is completely represented by the 
spike train it generates during that period of time. A neuronal spike train carries 
information in the following manner (Kandel, et al. 2000).  
(1) The number of action potentials (spikes); and 
(2) The time intervals between them. 
(Although this argument simplifies the description of brain functions to a great extent it 
ignores the reality of occurrence of change in the neural circuit without changing the 
firing patterns of the neurons. This has been discussed in Conclusion). The duration of a 
spike is typically 1 to 2 milliseconds (ms, depending on the temperature) (Koch 1999). 
When the sample frequency is high (1000 Hz or more) by FFT reliable information out of 
a neuronal spike train can be extracted. The FFT produces the vector 
bababaa ....
22110
where 
     
=na (nth  Fourier coefficient + conjugate of nth  Fourier coefficient),                (2.1) 
     
= (nth  Fourier coefficient – conjugate of nth  Fourier coefficient)                (2.2) 
1−=i . For convenience it can be rewritten as )12().....1( +ree , where 
     
− oddisnifbne
evenisnifane
,                                                                                       (2.3) 
in uniform symbol. More conventionally the vector kv  associated with the spike train of 
the kth  neuron in the network can be written as  
     
kkk reev ))12(),...,1(( += .                                                                                       (2.4) 
The suffix T  stands for transpose. If the duration of the spike train is p  seconds then 
pr 500=  (assuming sample frequency is 1000 Hz). Clearly the vector kv  uniquely 
represents the spike train of the kth  neuron in the network, assuming all the neurons in 
the network have been uniquely numbered. Since the Fourier series is convergent 
0)12( →+rek  as ∞→r . 
     Let there are N  neurons in the network. The behavior iB  of the network for a 
duration of p  seconds is represented by the cluster of N vectors { }N
kv 1= , where 
     
k reev ))12(),...,1(( += ,                                                                                       (2.5) 
     p
= ,                                                                                                                    (2.6) 
f  is sample frequency. It is important to note that even when a neuron is oscillating 
below threshold for spike initiation, it can still release neurotransmitter and shape the 
final circuit output (Harris-Warrick & Marder 1991). But for simplicity of modeling I 
shall ignore this fact in this paper. 
C) Memory 
     Memory is no single concept and there is no universally agreed upon definition of it. 
However the starting point for virtually any scientific analysis of memory involves a 
decomposition into processes of encoding, storage and retrieval  (Schacter 2004). Wilder 
Penfield explored the cortical surface in more than a thousand epileptic patients. On rare 
occasions (about 8% of all the subjects he tried) he found that electrical stimulation in the 
temporal lobes produced a coherent recollection of an earlier experience (Kandel et al. 
2000). A similar phenomenon has been observed by stimulating the inferotemporal (IT) 
cortex of macaque monkeys (Afraz et al. 2006). By mild stimulation to the IT in the 
macaque brain (previously trained to distinguish between face and non-face) impression 
of seeing a face was created in the mind of the animals where there was actually no face. 
An opposite phenomenon has been reported for the human brain (Quian Quiroga et al. 
2005). Visualization of images of interesting objects including faces of celebrities can 
make particular neurons to fire in the sub-region of the medial temporal lobes (MTL) of 
the human brain consisting of hippocampus, amygdala, entorhinal cortex and 
parahippocampal gyrus. This supports the hypothesis – object cognition activates a 
network in the brain and artificially activating the network to the appropriate degree will 
create the impression of perceiving the object in the brain even when it is not present in 
the environment. The following hypothesis seems to hold. 
A particular cognition involves a particular network in the brain where the memory of the 
cognition remains stored. The higher processing areas of the network takes increasingly 
greater part in the cognition and also greater part in storing and retrieving the memory. 
For the purpose of this paper it would be enough to be able to express memory in terms 
of behavior of a network. If the behavior of the network is iB  the memory associated 
with it will be { }N
ki uM 1== . The relationship between 
ku  and 
kv  is determined by long 
term potentiation (LTP) and long term depression (LTD) if iM  is residing in the network 
long after iB  has happened. It may be appropriate to emphasize at this point that the 
collective firing pattern of neurons (i.e., the collective spike trains of neurons belonging 
to the network) to evoke iM  is preserved by the synaptic connections in the network, for 
they control input to the neurons in response to the stimuli and therefore iM  is stored in 
the synaptic strengths of the network. A metric between iB  and iM  can be defined in the 
following manner 
     { }∑ ∑
MBd ii ,                                                          (2.7) 
where N  is the number of neurons in the network. Although virtually impossible to 
compute in this form d  can be a measure of plasticity of the whole network with respect 
to iB  and iM . Note that the time duration has been taken care of in p  at the time of 
determining )( je i
 and )( je i
 (neuronal firing patterns may not be identical at the time 
of a behavior and recalling it later both with respect to identical set of stimuli). A single 
network can mediate multiple behaviors (Harris-Warrick & Marder 1991) and therefore 
can store multiple memories. 
     Memory associated with a behavior may either be positive (appetitive) or negative 
(aversive) or neutral or a combination of them. The sense positive, negative and neutral 
are totally subjective and no attempt will be made here to define them. The meaning will 
become clear in the contexts in which they occur. A simple behavior is one with which 
only one type of memory (i.e., either positive or negative or neutral) remains associated. 
A behavior with combined types of memory can be called complex behavior and let us 
assume any behavior can be decomposed into simple behaviors. A network therefore can 
be thought to have memories of simple behaviors only. When the difference between iB  
and iM  is small i.e., in (2.7) ),( ii MBd  becomes small, consolidation of iM  is good. 
),( ii MBd  is intimately related to synaptic plasticity. 
D) Plasticity 
     (2.7) gives us an immediate measure (no matter however difficult it is or may even be 
impossible to implement) of plasticity of a neuronal network. The measure of combined 
long term plasticity )(tW  as a combination of long term potentiation (LTP) and long 
term depression (LTD) can be given by the following formula 
     ∑
ii MBd
),( ,                                                                                        (2.8) 
where p is the duration of recording of the neural response when iB  is happening and 
when iM  is being retrieved both in response to identical set of stimuli and t  is the time 
difference between end of happening of iB  and start of retrieving iM . Usually p  
remains fixed and therefore )(),( tWtpW = . m  is the total number of past behaviors 
whose memory is still preserved in the network. When the gap between occurrence of iB  
and recalling of iM  is long (30 minutes or more according to some estimate (Koch 
1999)) (2.8) gives the long term plasticity and when it is shorter (2.8) gives short term 
plasticity. The following notion will also be useful 
     ),()( iii MBdtW = ,                                                                                                   (2.9) 
where )(iWi  is the network plasticity between behavior iB  and memory iM  after time t . 
E) Feeling 
     From a modeling or computational point of view the feelings may be taken as 
mediating the intensity of a behavioral response. In this sense if iB  is a simple behavior 
the feeling associated with it determines how intensely negative or intensely positive will 
be the memory of it. Ideally iM  should have an ‘intensity distribution function’ similar 
to a normal distribution function, which is decayed as the distance from the mean 
position is increased. iM  is not a single point, but a cluster of points. So if the intensity 
distribution function is to be modeled after the normal distribution function the most 
appropriate candidate for the mean point turns out to be the mean point of iM  
     
M ii 
+= ∑∑
)12(),.......,1(
,                                                           (2.10) 
where T  stands for transpose. The feeling function RRF
:  associated with iM  is 
to be defined as 
     
2/12/)12(
i MXMXXF
,                               (2.11) 
where Σ  is the covariance matrix and Σ  is the determinant of Σ . Since X  has 12 +r  
independent component variables only diagonal entries of Σ  are nonzero and each of 
them is the variance of a component variable in X . Σ  is the parameter which controls 
the intensity of )(XFi  beyond iM . The hormones and neuromodulators responsible for 
mediating feelings act by controlling the entries of Σ . 
F) Interaction 
     Interaction means here the part played by synapses in the network in mediating the 
effects of sM i '  on an ongoing behavior B . In mathematical term it can be put as 
     ))(())(()(
tpWfttpWftI iii −−+−= ,                                                                 (2.12) 
where )(
tI i  denotes the interaction and t  is any instant during occurrence of B . if  is a 
continuous function which is almost everywhere differentiable. When B  takes place the 
plasticity of the network changes and so also the interaction. Let us normalize interaction 
by the following formula 
     
)( ,                                                                                                   (2.13) 
where T  is the duration of happening of B . 
3. Integrative dynamics 
     Now let us briefly consider a phenomenon in electrostatics. Let there be seven charged 
particles on a plane and they are all in arbitrary but fixed positions. Four of them are 
positively charged and three are negative. A new negative charge is introduced, which is 
allowed to move freely in the plane, that is, it has two degrees of freedom along the X 
and Y axes. Assume apart from sign all the charges are quantitatively equal. 
     Let the locus of the introduced negative charge be ),( yx . The position of four positive 
charges be { }4
),( =iii yx  and that of the three negative charges be { }
),( =iii yx . The 
Coulomb force ),( yxF  acting on the free charge is given by 
     ∑∑
== −+−
)()()()(
i iii ii yyxx
yxf ,                          (3.1) 
where C  is a constant. (3.1) will govern the dynamics of the whole system. Now what 
should be the condition to keep the introduced charge fixed within a bounded region 
enclosing the seven fixed positioned charge so that the potential energy on the introduced 
particle becomes minimum? 
     On line of this analogy the dynamics of a new behavior B  should follow the 
governing expression )(BG  given by the following equation 
     
( ) ( )
= +−=
smi i
)( ,                                                    (3.2) 
where m  is the number of simple behaviors whose memories are stored in the network 
under consideration. It has been assumed that B  will be attracted towards s  behaviors 
with positive experience and will be repulsed by the remaining sm −  behaviors with 
negative experience. Like (3.1) there is no apparent reason why inverse square law 
should hold for (3.2) also. If the inverse square law does not hold then the denominators 
on the right side of (3.2) should be replaced by a general polynomial in the metric d , for 
any continuous function can be approximated to any desired degree within a compact 
interval by a suitable polynomial. Like the electrostatic situation for a given iM  )( iMG  
must have a unique pole at the iM . This means in general (3.2) can have the form 
      
( ) ( )
= +−=
1 1 ),(
)( ,                                                   (3.3) 
for some positive n . However in this paper I shall adhere to (3.2), for the model here is 
essentially electrophysiological and therefore Coulomb interaction seems to be probable. 
In summary the dynamics of the new behavior B  is described by (2.7), (2.11), (2.13) and 
(3.2). 
     In analogy with the electrostatic system (3.2) is a ‘force like’ expression with which a 
‘potential energy like’ expression needs to be associated, which for the sake of stability 
of the system (i.e., if perturbed infinitesimally will come back to the original state) needs 
to be minimized. Let the potential function associated with B  due to iM  be 
iφ  given by 
     ∑
))12(),......,1((φφ .                                                                           (3.4) 
The potential function associated with the emerging new behavior B  due to all previous 
behaviors iB  ( iM  is memory of iB ) be φ , which is given by 
     ∑
φφ .                                                                                                                (3.5) 
To derive the potential energy classically from the force field the force must have to be 
conservative, i.e., independent of time (Goldstein 1950). In (3.2) the expression )(tI i  is a 
function in time. However in case of a very strong aversive stimulus like, locating a 
predator dangerously close all of a sudden or being on the verge of falling down deep 
underneath from a very high roof top after a sudden slip, an extremely fast behavioral 
response must have to be shown. Within this duration synaptic plasticity does not get 
much chance to act and the feeling must have to be very strong (such as intense fear) to 
compensate for that as is evident from (3.2). This gives us an opportunity to treat )(tI  as 
a fixed quantity which makes )(BG  in (3.2) time independent for the duration of B . 
Then in analogy with the electrostatic system )(BG  must satisfy the following relation 
     ∑ ∑∑
.                                                                       (3.6) 
The value of )(BG  has only been considered and not its direction. (3.2) and (3.6) 
together give 
     
( ) ( )
∑ ∑∑ ∑∑
= +−== =
smi i
jeN 1 1
.                 (3.7) 
(3.7) describes the dynamics of the behavior of B  in the 
12 +r
R  under the assumption that 
)(tI i  is constant (otherwise the ‘force field’ would not have been conserved and only 
space dependent potential expressions could not have been brought in the dynamics). 
Also the time scale is very small. φ  as given by (3.5) will have to be minimized 
(minimization of energy is an important criterion for neural computation (Laughlin 
2004)), which means 
     )}12(,.....,1{,0
.                                                                          (3.8) 
Combining (3.4) and (3.5) the expression for φ  becomes 
     ∑∑
k ree
))12(),......,1((φφ .                                                                        (3.9) 
If (3.9) is to give a global minimum it should not only be true when φ  is given by (3.9), 
but also it must hold when φ  is given by 
     ∑ ∑
kki reeyx
))12(),......,1((φφ ,                                                                (3.10) 
where )1,1(, ccyx ki +−∈  for some small 0>c . This is because a global minimum is a 
very stable position and therefore under a small perturbation the system always comes 
back to it. Let ikki zyx = . Then combining (3.8) and (3.10) the following is obtained 
     
+++ 0
.....
.....
.....
.....
.....
.....
.....
12,,12,2,112,1,1
2,,2,2,12,1,1
1,,1,2,11,1,1
NmrNmrr
,                                                (3.11) 
where 
))12(),......,1((
. Note that each kiz ,  or ikz  can take 
uncountably infinitely many values from some (small) open interval and for all of them 
(3.11) holds under the perturbation principle. Therefore the mNr ×+ )12(  matrix on the 
left of (3.11) must represent a null linear transformation and it must be a null matrix, 
which implies 
     jki
jki ,,,0
))12(),......,1((
,, ∀=
.                                                      (3.12) 
(3.7) and (3.12) together imply 
     
( ) ( )
= +−=
smi i
.                                                                   (3.13) 
(3.13) says, “Negative and positive experiences in a neuronal network must counter 
balance each other for a stable (which will not be altered due to presence of some amount 
of noise or distraction) unsupervised learning from the experience gained through a new 
behavior.” 
4. Application 
     What do we get from the principle enunciated at the end of the last section in response 
to a strong aversive stimulus? Its mathematical formulation (3.13) says, in the face of 
strong aversive stimuli (such as an imminent danger) the ensuing behavior B  must avoid 
repeating the past behaviors sBi '  with negative experience sM i ' . Note that the stimuli 
can invoke an iM  if and only if iB  is at least partially activated by the stimuli. (3.13) 
says the ensuing behavior B  will have to be such that none of the sBi '  with negative 
experience is repeated. This means in the information space 
12 +r
R  B  will have to sit 
away from each 
iM , which denotes memory of a negative behavior. B  will have its 
own positive and negative parts which will later be stored as new positive and negative 
memories in the network. B  cannot be neutral. This will be the subject of a future work. 
A) Aversion response latency 
     The prefrontal cortex participates in linking perception of stimuli to the guidance of 
behavior including the flexible execution of strategies for obtaining rewards and avoiding 
punishments as an organism interacts with its environment. Recording from neurons 
within healthy tissue in the ventral sites of the right prefrontal cortex short latency (120–
160ms) responses selective for aversive visual stimuli have been observed. No such 
latency was observed for pleasant or neutral stimuli (Kawasaki et al. 2001). (3.13) can 
offer us an explanation of this phenomenon. Aversive stimuli do evoke negative (or 
aversive) memory (that is how the stimuli are identified as aversive, even a novel 
aversive stimulus will have to be decomposable into known aversive features) and 
therefore the ensuing behavior B  must ensure that it has minimum overlap with the 
negative memories in the space 
12 +r
R . This in turn makes sure that B  acts on the 
network as less as possible to repeat the behaviors associated with the negative memories. 
     When there will be only aversive stimuli (no pleasant stimulus) the new behavior will 
have to be organized to avoid the stimuli particularly when the stimuli are strongly 
aversive (like an immediate threat). Clearly a strongly aversive stimulus must invoke the 
memory  
jM  of at least one simple negative behavior with which a strong feeling is to 
be associated. In other words amygdale make sure that the Σ  in (2.11) have larger 
entries. Then there must be the memory 
jM  of a simple positive behavior which can 
take appropriate action in response to the behavior (sensation) corresponding to 
jM . 
Even a new born is hardwired to express displeasure by crying in response to an aversive 
stimulus so that some one else (possibly the mother) is alerted and come in help to avoid 
the stimulus. In order to avoid the aversive stimuli the dynamics of B  will be such that 
(3.13) can take the following form 
     
( ) ( )∑∑ == +
,                                                                          (4.1) 
where )(XF j  is the feeling associated with 
jM  and )(tI j  is the network interaction 
between B  and 
jM , k  is the number of simple positive behavior recalled. This will 
counterbalance the aversive effect of the stimuli and make sure that (3.13) holds. 
     Whereas in case of pleasant or neutral stimuli (without the presence of aversive 
stimuli) avoidance is not necessary, the brain is free to repeat the behaviors with pleasant 
memory. In that case no global minimization of the potential function will be necessary 
and left side of (3.7) does not need to be zero. Therefore equality in (4.1) will also not be 
necessary. The new behavior can sit anywhere and no ‘optimum positioning’ (making the 
potential function globally minimum) will be necessary and this will require less time for 
a response and therefore there will be no latency in the response to pleasant stimuli. 
     When there will be both positive and negative stimuli (3.13) will hold. Thus it appears 
that whenever aversive (negative) stimuli are present either (3.13) or (4.1) needs to be 
calculated. Time taken by this ‘calculation’ in the brain is the reason for latency when 
there are aversive stimuli. Whereas no such latency is necessary for positive or neutral 
stimuli. Probably the brain does not calculate the way shown in this paper. Had it done so 
it would have taken a much longer latency. But the reasoning here is compelling enough 
to conclude that calculations responsible for the observed latency do take place in the 
brain in one form or the other. 
B) Sequential decision making 
     A brain always has to make choices i.e., a behavior is a series of switching or decision 
making procedures. Rabinovich and colleagues have considered the dynamics of decision 
making (DM) by the brain or cognitive state machine (CSM) at the psychological level 
(Rabinovich et al. 2006). Whereas the focus of this paper is on the dynamics of neural 
substrate of such processes. At each DM instant of a CSM the underlying neural 
dynamics of the DM generates the psychological process of DM in the CSM. 
     The dynamics of the CSM is governed by the Lotka-Volterra type equation 
(Rabinovich et al. 2006) 
     )(),( taatIaa i
jijiiii ηρσ +
+−= ∑
& ,                                                             (4.2) 
where )(tai  is the state of the CSM, ),( tIiσ  is a control function which controls the 
dynamics given by (4.3), I  is the input (environmental stimulus), ijρ  is a coupling 
constant between ia  and ja  based on genetic and memorized information (very similar 
to the interaction )(tI i  (2.13) between the memories of the old behavior and the ensuing 
new behavior, which in this paper has been taken to be constant), N  the total number of 
states and )(tiη  is the external noise. Note that in (4.2) the meaning of iσ  and N  are 
different than anywhere else in the earlier part of this paper. 
     
& ,                                                                                                  (4.3) 
where iU  is a potential function and τ  is characteristic time which is very small. 
Difference in states occur at the it  where a decision needs to be made as shown in Figure 
1. ia  can be chosen at an instant jt , where a decision has to be made, as many ways as 
there are minima of iU . 
                             
Figure 1: A sequence of cognitive states. Thin lines are possible paths and the decision 
path has been shown by thick lines. 9643 ,,, tttt  are the instances to make a decision. 
Adopted from (Rabinovich et al. 2006). 
     (3.13) is concerned about the dynamics of neural substrate of each single decision 
making (DM) in the sequence of DM’s shown in Figure 1. (3.13) is also derived by 
minimizing the potential function (3.9). Since the dynamics given by (3.7) is concerned 
about a local decision making in Figure 1, it was possible to take global optimum of the 
potential function given by (3.9) (global optimum of local potential function) and its 
stability ensures that the local dynamics is not perturbed by a small amount of noise. It is 
very important for stable cognition or stable behavior. According to this model the 
alteration in behavior (to make it dynamic) should be introduced by changing feeling 
(given by (2.11)) and synaptic plasticity (given by (2.13)), which happens from state to 
state in a CSM. Therefore despite the global minimum of the local dynamics the CSM 
manages to move on from one state to the next till it reaches the end of life which may be 
signified by lack of emotional impetus. On the other hand at the sequential DM level in a 
CSM (Figure 1) introduction of noise plays a significant role in dynamically altering the 
behavior (equation (4.2)). In both the dynamical systems of this paper and that of 
Rabinovich et al. the time length has been taken to be short. The duration has been taken 
so short that the dynamics of ),( tIiσ  (in (4.3)) and )(tI i  (in (2.13)) do not change 
within that period. In this scenario comparing (3.13) and (4.2) it appears that the external 
noise )(tiη  may take an important role in mediating feeling (emotional distraction or 
attraction with respect to an external stimulus). 
     (3.13) and (4.1) describe how new behavior is formed depending on synaptic 
plasticity, memory and feeling. However there is no deterministic way to compute the 
new behavior from (3.13) or (4.1) even if all the information are available. Also (3.13) or 
(4.1) will be extremely difficult (if not impossible) to compute except for cases like a 
simple behavior (such as inking or gill withdrawal) of a simple animal (such as the sea 
snail Aplysia californica). This is in conformity with the fact that closer is the model to 
the neuronal network level of the brain the more difficult it would be to implement. On 
the other hand the CSM dynamics given by (4.2) is very convenient to compute. 
C) Potential function 
     The potential function introduced by (3.9) in the neuronal network dynamics of the 
brain can be related to metabolic energy consumption during activation of the network to 
execute a behavioral or cognitive task. (3.9) is based on electrophysiology and 
metabolism is related to hemodynamics. Lot of research is going on to find the 
relationship between these two activities of the brain (Logothetis & Wandell 2004). 
Recently a correlation study between stimulus based ERP and fMRI in different parts of 
the human brain has been reported (Gore et al. 2006). Despite convincing evidence of 
their interdependence, such as in the form of dependence of fMRI on ERP or EEG, 
precise knowledge about the nature of dependence is still lacking. A linear transform 
model had been proposed based on the hypothesis that fMRI responses are proportional 
to local average neural activity averaged over a period of time (Boynton et al. 1996). It 
has been reported that in the visual cortex of macaque monkey fMRI response depends 
closely on the local field potential (LFP) (Logothetis et al. 2001). 
     Note that the potential function given by (3.5) can be obtained from (3.7) when the 
time course is short and the quantities on the right side are known. From (3.7) it is clear 
that the potential function φ  (which is defined on the Fourier coefficients of neural spike 
trains (3.9)) depends on past memory, synaptic plasticity and feelings. In case of a 
potential function φ  sum of local components will give the function over the whole 
space, and therefore no matter how large and distributed the network is, local component 
of φ  at a point x  on the cortex (let be denoted by xφ ) can be represented by (3.7) where 
only the neurons in a small neighborhood of x , denoted by )(xn , will participate in the 
computation of xφ . Since xφ  defined on the Fourier coefficients of spike trains of the 
neurons in )(xn  and the value of xφ  depends on past memories, plasticity of synapses 
within )(xn  and part of feeling arising within )(xn  it is supposed to have a role in BOLD 
fMRI response at x  with respect to a stimulus. xφ  may have an affine or linear relation 
with stimulus driven difference in local BOLD signal at x  as modeled in (Logothetis & 
Wandell 2004). But this needs experimentations to establish. 
Conclusion 
     In this paper a system level model of the brain functions for behavior has been 
proposed. The model needs input from individual neurons under the assumption that a 
brain circuit responsible for a behavior can be understood by the behavior of neurons 
alone. This assumption has limitations, for significant change in a neural network may 
occur at the spiking sub-threshold level in the neurons and therefore without changing the 
spike trains (Harris-Warrick & Marder 1991). Note that FFT based information retrieveal 
technique followed in this paper will hold equally good for subthreshold signals, whereas 
spike detection or prediction algorithms may not work for those signals. To account for a 
fuller neural computation synaptic computation (Abbott & Regehr 2004) and dendritic 
computation (London & Hausser 2005) will also have to be incorporated. A close 
investigation into the nature of )(tI  (equation (2.13)) will inevitably call attention to 
synaptic and dendritic computations. The model equation (3.2) will be valid in all 
generality, but the important consequence (3.13) cannot be drawn as easily as shown in 
this paper. 
     The effectiveness of the model dynamics is only for a very short period of time and 
therefore only behaviors in response to strong aversive stimuli have been considered, for 
such response behaviors must have to be very quick. The time duration is so short that it 
has been assumed – the average change in the collection of synapses in a network 
remains fixed within that time. This is an important constraint for introducing potential 
energy function whose global minimization gives stability to the behavioral response in 
the sense that it remains unperturbed in a noisy environment. 
     Even without incorporating synaptic and dendritic computations the model stands 
extremely difficult to implement (for example, calculating iB  and then calculating iM  
will be very challenging). However it can be implemented in case of a simple behavior by 
a simple animal. In invertebrates like the sea snail Aplysia lesser number of neurons will 
have to be monitored (only a few in case of simple behaviors like gill withdrawal or 
inking) and single cell recordings will be less challenging than the mammals. This will 
make the verification of the system possible. In case of human brain functions if the 
behavior of the circuit involved can be monitored to a good extent by recording signals 
only from a few neurons verification of the model may be feasible for some very simple 
behavior like the following. Consider the experiment of tapping the leg (Kandel et al. 
2000). If there is a sharp edge such that dragging the leg too much backward will make it 
bump on the sharp edge which is another aversive stimulus. The final position of the leg 
will be away from the source of tapping as well as the sharp edge. 
     If a relationship between the potential function φ  and the metabolic or hemodynamic 
activity of the brain can be established, study of the system will become easier. 
Introduction of an electrophysiological potential function in the neural computation is a 
significant outcome of the FFT based information extraction method followed in this 
paper. Apart from the prospect of relating it to metabolic energy requirements of the 
brain φ  gives the opportunity of deriving interesting theoretical results as shown in this 
paper. 
     Since the expression of feeling as given by (2.11) is independent of time here the 
detail of feeling did not have to be considered. But feeling is likely to be dependent on 
time in the long run and in that case the entries of Σ  (which is a diagonal matrix) will be 
function of time and the dynamics will be much more complicated. How the eigen values 
of Σ  are controlled by the limbic system of the brain is an open question. No attempt has 
been made in this paper to answer the question. 
Acknowledgement 
     The author thankfully acknowledges the Institute of Mathematical Sciences in 
Chennai, India for a postdoctoral fellowship under which this work has been carried out. 
Some comments by Peter Dayan on a preliminary version of this manuscript are also 
being acknowledged. 
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  In this paper a theoretical model of functioning of a neural circuit during a
behavioral response has been proposed. A neural circuit can be thought of as a
directed multigraph whose each vertex is a neuron and each edge is a synapse.
It has been assumed in this paper that the behavior of such circuits is
manifested through the collective behavior of neurons belonging to that
circuit. Behavioral information of each neuron is contained in the coefficients
of the fast Fourier transform (FFT) over the output spike train. Those
coefficients form a vector in a multidimensional vector space. Behavioral
dynamics of a neuronal network in response to strong aversive stimuli has been
studied in a vector space in which a suitable pseudometric has been defined.
The neurodynamical model of network behavior has been formulated in terms of
existing memory, synaptic plasticity and feelings. The model has an analogy in
classical electrostatics, by which the notion of force and potential energy has
been introduced. Since the model takes input from each neuron in a network and
produces a behavior as the output, it would be extremely difficult or may even
be impossible to implement. But with the help of the model a possible
explanation for an hitherto unexplained neurological observation in human brain
has been offered. The model is compatible with a recent model of sequential
behavioral dynamics. The model is based on electrophysiology, but its relevance
to hemodynamics has been outlined.

Introduction 
     Quick behavioral response to strong aversive stimuli (such as threat from a predator or 
an imminent danger of being hurt) is a key to survival throughout the animal kingdom. 
Network models of animal behavior have been elaborately discussed in (Schmajuk 1997). 
A neural network model of rats’ anxiety behavior had been studied by Salum and 
colleagues (Salum et al. 2000), but they didn’t take into account the functioning of the 
nerve cells in the brain during the task performance. Recently a neurodynamical model 
for conditional visuomotor association task has been proposed (Loh & Deco 2005). In 
this model a trial and error paradigm has been assumed in a stochastic decision space. An 
integrate and fire neuronal network model has been proposed to realize the paradigm. A 
neural network model of brain or cognitive state machine (CSM) to study decision 
making in a competitive environment has also been proposed (Rabinovich et al. 2006). 
The dynamics of making a choice from among multiple conflicting options has been 
formulated by Lotka-Volterra type of equations. It is evident that intelligent decisions in a 
sequential behavior have to be stable against noise and reproducible to allow 
memorization and reuse of successful decision sequences in the future. On the other 
hand, it also has to be sensitive to new information from the environment. These two 
fundamentally contradictory requirements have been taken care of in (Rabinovich et al. 
2006). 
     In this paper a neurodynamical model of response behavior of a neural network to 
strong aversive stimuli has been presented. The assumption is that the network will 
behave in a manner to avoid repeating the negative experiences of the past. No specific 
neuronal network model has been assumed. In this approach information has been 
extracted (at least theoretically) from a behaving neuronal network by FFT on spike 
trains of all the neurons in the network, which should work across all networks of 
neurons irrespective of their architecture. The neurodynamical system has a unique 
representation in  the information space where the Fourier coefficients of a spike train are 
arranged as a vector and uniquely represent the spike train. All the calculations have been 
carried out in that space. The model depends on past memory, synaptic plasticity and 
intensity of feeling. Interestingly, a short latency (120 – 160 ms) had been reported before 
response to aversive stimuli in the right prefrontal cortex of a human subject. No such 
latency was observed in case of pleasant or neutral stimuli (Kawasaki et al. 2001). The 
present model can offer a possible explanation for this apparently perplexing 
phenomenon. The following assumptions have been made: 
1) Structure: Any behavior involves a network in the nervous system which if 
represented as a directed multigraph will have neurons as vertices and synapses as 
(directed) edges. 
2) Function: The functional behavior of this network is manifested by the collective 
behavior of the neurons present in the network. 
3) Memory: Memory of experiences of past behaviors of this network is stored 
entirely within the network and nowhere out side of it. 
4) Plasticity: Plasticity of each synapse is a time dependent function (called synaptic 
weight) and the total plasticity of the network between any two behaviors is also a 
time dependent function called network plasticity. 
5) Feeling: Feeling is associated with each behavior as a specific mathematical 
function which controls how experience associated with this behavior will mediate 
the intensity of effect of this behavior on any other behavior. 
6) Interaction: Network plasticity mediates the interaction of the ongoing behavior 
with the memory of past behaviors stored in the network. 
In section 2 each of the above points will be explained briefly and will be represented 
with appropriate mathematical expressions and equations. In section 3 with the help of an 
analogy from classical electrostatics the behavior dynamics of the network in terms of 
those expressions and equations will be formulated. Notion of a force like and a potential 
energy like expressions have been introduced. Despite computational difficulties 
involved with the model, in section 4 it has been related to reality first by offering an 
explanation to an hitherto unexplained observation in the human brain, second relating it 
with a successful decision making model and third relating this electrophysiological 
model to hemodynamical activity of the brain. The paper will be concluded with 
discussions and future directions. 
2. Modeling of the parts 
A) Structure 
     The closest computational analog of a neuronal circuit is a directed multigraph, whose 
each node will represent a neuron and each edge a synapse (Majumdar & Kozma, 2006). 
It can be represented as a three dimensional array ],,[ kjia . If there are p  different 
synapses joining the neuron i  with the neuron j  (assuming that all neurons in the brain 
have been numbered) then ],,[ kjia  will give the weight (the gain in synaptic 
transmission) of the kth  synapse joining the neuron i  with the neuron j , where 
pk ≤≤1 . If the neuron i  and neuron j  are excitatory then ],,[ kjia  will be positive, if 
they are inhibitory then ],,[ kjia  will be negative. For given values of ji,  and k  ],,[ kji  
uniquely determines a synapse and ],,[ kjia  represents the synaptic weight. In general 
],,[ kjia  will be a function of time and may be written as )](,,[ tkjia  or )(taijk , where t  
denotes time. 
B) Function 
     Computations in the central nervous system (CNS) have been viewed from three 
different angles – (a) synaptic computation (Abbott & Regehr 2004), (b) dendritic 
computation (London & Hausser 2005) and (c) neuronal computation (Koch 1999; 
Borisyuk & Rinzel 2005). In a neuronal network synapses are activating the neurons and 
neurons are activating the synapses. In this sense the aggregate behavior of the neurons in 
a network is equivalent to aggregate behavior of the synapses in that network. On an 
average every neuron receives input from about 
10  synapses and therefore from the 
computational modeling purpose it would be more convenient to consider the aggregate 
behavior of neurons rather than the aggregate behavior of synapses in a network. The 
behavior of a neuron during a particular epoch of time is completely represented by the 
spike train it generates during that period of time. A neuronal spike train carries 
information in the following manner (Kandel, et al. 2000).  
(1) The number of action potentials (spikes); and 
(2) The time intervals between them. 
(Although this argument simplifies the description of brain functions to a great extent it 
ignores the reality of occurrence of change in the neural circuit without changing the 
firing patterns of the neurons. This has been discussed in Conclusion). The duration of a 
spike is typically 1 to 2 milliseconds (ms, depending on the temperature) (Koch 1999). 
When the sample frequency is high (1000 Hz or more) by FFT reliable information out of 
a neuronal spike train can be extracted. The FFT produces the vector 
bababaa ....
22110
where 
     
=na (nth  Fourier coefficient + conjugate of nth  Fourier coefficient),                (2.1) 
     
= (nth  Fourier coefficient – conjugate of nth  Fourier coefficient)                (2.2) 
1−=i . For convenience it can be rewritten as )12().....1( +ree , where 
     
− oddisnifbne
evenisnifane
,                                                                                       (2.3) 
in uniform symbol. More conventionally the vector kv  associated with the spike train of 
the kth  neuron in the network can be written as  
     
kkk reev ))12(),...,1(( += .                                                                                       (2.4) 
The suffix T  stands for transpose. If the duration of the spike train is p  seconds then 
pr 500=  (assuming sample frequency is 1000 Hz). Clearly the vector kv  uniquely 
represents the spike train of the kth  neuron in the network, assuming all the neurons in 
the network have been uniquely numbered. Since the Fourier series is convergent 
0)12( →+rek  as ∞→r . 
     Let there are N  neurons in the network. The behavior iB  of the network for a 
duration of p  seconds is represented by the cluster of N vectors { }N
kv 1= , where 
     
k reev ))12(),...,1(( += ,                                                                                       (2.5) 
     p
= ,                                                                                                                    (2.6) 
f  is sample frequency. It is important to note that even when a neuron is oscillating 
below threshold for spike initiation, it can still release neurotransmitter and shape the 
final circuit output (Harris-Warrick & Marder 1991). But for simplicity of modeling I 
shall ignore this fact in this paper. 
C) Memory 
     Memory is no single concept and there is no universally agreed upon definition of it. 
However the starting point for virtually any scientific analysis of memory involves a 
decomposition into processes of encoding, storage and retrieval  (Schacter 2004). Wilder 
Penfield explored the cortical surface in more than a thousand epileptic patients. On rare 
occasions (about 8% of all the subjects he tried) he found that electrical stimulation in the 
temporal lobes produced a coherent recollection of an earlier experience (Kandel et al. 
2000). A similar phenomenon has been observed by stimulating the inferotemporal (IT) 
cortex of macaque monkeys (Afraz et al. 2006). By mild stimulation to the IT in the 
macaque brain (previously trained to distinguish between face and non-face) impression 
of seeing a face was created in the mind of the animals where there was actually no face. 
An opposite phenomenon has been reported for the human brain (Quian Quiroga et al. 
2005). Visualization of images of interesting objects including faces of celebrities can 
make particular neurons to fire in the sub-region of the medial temporal lobes (MTL) of 
the human brain consisting of hippocampus, amygdala, entorhinal cortex and 
parahippocampal gyrus. This supports the hypothesis – object cognition activates a 
network in the brain and artificially activating the network to the appropriate degree will 
create the impression of perceiving the object in the brain even when it is not present in 
the environment. The following hypothesis seems to hold. 
A particular cognition involves a particular network in the brain where the memory of the 
cognition remains stored. The higher processing areas of the network takes increasingly 
greater part in the cognition and also greater part in storing and retrieving the memory. 
For the purpose of this paper it would be enough to be able to express memory in terms 
of behavior of a network. If the behavior of the network is iB  the memory associated 
with it will be { }N
ki uM 1== . The relationship between 
ku  and 
kv  is determined by long 
term potentiation (LTP) and long term depression (LTD) if iM  is residing in the network 
long after iB  has happened. It may be appropriate to emphasize at this point that the 
collective firing pattern of neurons (i.e., the collective spike trains of neurons belonging 
to the network) to evoke iM  is preserved by the synaptic connections in the network, for 
they control input to the neurons in response to the stimuli and therefore iM  is stored in 
the synaptic strengths of the network. A metric between iB  and iM  can be defined in the 
following manner 
     { }∑ ∑
MBd ii ,                                                          (2.7) 
where N  is the number of neurons in the network. Although virtually impossible to 
compute in this form d  can be a measure of plasticity of the whole network with respect 
to iB  and iM . Note that the time duration has been taken care of in p  at the time of 
determining )( je i
 and )( je i
 (neuronal firing patterns may not be identical at the time 
of a behavior and recalling it later both with respect to identical set of stimuli). A single 
network can mediate multiple behaviors (Harris-Warrick & Marder 1991) and therefore 
can store multiple memories. 
     Memory associated with a behavior may either be positive (appetitive) or negative 
(aversive) or neutral or a combination of them. The sense positive, negative and neutral 
are totally subjective and no attempt will be made here to define them. The meaning will 
become clear in the contexts in which they occur. A simple behavior is one with which 
only one type of memory (i.e., either positive or negative or neutral) remains associated. 
A behavior with combined types of memory can be called complex behavior and let us 
assume any behavior can be decomposed into simple behaviors. A network therefore can 
be thought to have memories of simple behaviors only. When the difference between iB  
and iM  is small i.e., in (2.7) ),( ii MBd  becomes small, consolidation of iM  is good. 
),( ii MBd  is intimately related to synaptic plasticity. 
D) Plasticity 
     (2.7) gives us an immediate measure (no matter however difficult it is or may even be 
impossible to implement) of plasticity of a neuronal network. The measure of combined 
long term plasticity )(tW  as a combination of long term potentiation (LTP) and long 
term depression (LTD) can be given by the following formula 
     ∑
ii MBd
),( ,                                                                                        (2.8) 
where p is the duration of recording of the neural response when iB  is happening and 
when iM  is being retrieved both in response to identical set of stimuli and t  is the time 
difference between end of happening of iB  and start of retrieving iM . Usually p  
remains fixed and therefore )(),( tWtpW = . m  is the total number of past behaviors 
whose memory is still preserved in the network. When the gap between occurrence of iB  
and recalling of iM  is long (30 minutes or more according to some estimate (Koch 
1999)) (2.8) gives the long term plasticity and when it is shorter (2.8) gives short term 
plasticity. The following notion will also be useful 
     ),()( iii MBdtW = ,                                                                                                   (2.9) 
where )(iWi  is the network plasticity between behavior iB  and memory iM  after time t . 
E) Feeling 
     From a modeling or computational point of view the feelings may be taken as 
mediating the intensity of a behavioral response. In this sense if iB  is a simple behavior 
the feeling associated with it determines how intensely negative or intensely positive will 
be the memory of it. Ideally iM  should have an ‘intensity distribution function’ similar 
to a normal distribution function, which is decayed as the distance from the mean 
position is increased. iM  is not a single point, but a cluster of points. So if the intensity 
distribution function is to be modeled after the normal distribution function the most 
appropriate candidate for the mean point turns out to be the mean point of iM  
     
M ii 
+= ∑∑
)12(),.......,1(
,                                                           (2.10) 
where T  stands for transpose. The feeling function RRF
:  associated with iM  is 
to be defined as 
     
2/12/)12(
i MXMXXF
,                               (2.11) 
where Σ  is the covariance matrix and Σ  is the determinant of Σ . Since X  has 12 +r  
independent component variables only diagonal entries of Σ  are nonzero and each of 
them is the variance of a component variable in X . Σ  is the parameter which controls 
the intensity of )(XFi  beyond iM . The hormones and neuromodulators responsible for 
mediating feelings act by controlling the entries of Σ . 
F) Interaction 
     Interaction means here the part played by synapses in the network in mediating the 
effects of sM i '  on an ongoing behavior B . In mathematical term it can be put as 
     ))(())(()(
tpWfttpWftI iii −−+−= ,                                                                 (2.12) 
where )(
tI i  denotes the interaction and t  is any instant during occurrence of B . if  is a 
continuous function which is almost everywhere differentiable. When B  takes place the 
plasticity of the network changes and so also the interaction. Let us normalize interaction 
by the following formula 
     
)( ,                                                                                                   (2.13) 
where T  is the duration of happening of B . 
3. Integrative dynamics 
     Now let us briefly consider a phenomenon in electrostatics. Let there be seven charged 
particles on a plane and they are all in arbitrary but fixed positions. Four of them are 
positively charged and three are negative. A new negative charge is introduced, which is 
allowed to move freely in the plane, that is, it has two degrees of freedom along the X 
and Y axes. Assume apart from sign all the charges are quantitatively equal. 
     Let the locus of the introduced negative charge be ),( yx . The position of four positive 
charges be { }4
),( =iii yx  and that of the three negative charges be { }
),( =iii yx . The 
Coulomb force ),( yxF  acting on the free charge is given by 
     ∑∑
== −+−
)()()()(
i iii ii yyxx
yxf ,                          (3.1) 
where C  is a constant. (3.1) will govern the dynamics of the whole system. Now what 
should be the condition to keep the introduced charge fixed within a bounded region 
enclosing the seven fixed positioned charge so that the potential energy on the introduced 
particle becomes minimum? 
     On line of this analogy the dynamics of a new behavior B  should follow the 
governing expression )(BG  given by the following equation 
     
( ) ( )
= +−=
smi i
)( ,                                                    (3.2) 
where m  is the number of simple behaviors whose memories are stored in the network 
under consideration. It has been assumed that B  will be attracted towards s  behaviors 
with positive experience and will be repulsed by the remaining sm −  behaviors with 
negative experience. Like (3.1) there is no apparent reason why inverse square law 
should hold for (3.2) also. If the inverse square law does not hold then the denominators 
on the right side of (3.2) should be replaced by a general polynomial in the metric d , for 
any continuous function can be approximated to any desired degree within a compact 
interval by a suitable polynomial. Like the electrostatic situation for a given iM  )( iMG  
must have a unique pole at the iM . This means in general (3.2) can have the form 
      
( ) ( )
= +−=
1 1 ),(
)( ,                                                   (3.3) 
for some positive n . However in this paper I shall adhere to (3.2), for the model here is 
essentially electrophysiological and therefore Coulomb interaction seems to be probable. 
In summary the dynamics of the new behavior B  is described by (2.7), (2.11), (2.13) and 
(3.2). 
     In analogy with the electrostatic system (3.2) is a ‘force like’ expression with which a 
‘potential energy like’ expression needs to be associated, which for the sake of stability 
of the system (i.e., if perturbed infinitesimally will come back to the original state) needs 
to be minimized. Let the potential function associated with B  due to iM  be 
iφ  given by 
     ∑
))12(),......,1((φφ .                                                                           (3.4) 
The potential function associated with the emerging new behavior B  due to all previous 
behaviors iB  ( iM  is memory of iB ) be φ , which is given by 
     ∑
φφ .                                                                                                                (3.5) 
To derive the potential energy classically from the force field the force must have to be 
conservative, i.e., independent of time (Goldstein 1950). In (3.2) the expression )(tI i  is a 
function in time. However in case of a very strong aversive stimulus like, locating a 
predator dangerously close all of a sudden or being on the verge of falling down deep 
underneath from a very high roof top after a sudden slip, an extremely fast behavioral 
response must have to be shown. Within this duration synaptic plasticity does not get 
much chance to act and the feeling must have to be very strong (such as intense fear) to 
compensate for that as is evident from (3.2). This gives us an opportunity to treat )(tI  as 
a fixed quantity which makes )(BG  in (3.2) time independent for the duration of B . 
Then in analogy with the electrostatic system )(BG  must satisfy the following relation 
     ∑ ∑∑
.                                                                       (3.6) 
The value of )(BG  has only been considered and not its direction. (3.2) and (3.6) 
together give 
     
( ) ( )
∑ ∑∑ ∑∑
= +−== =
smi i
jeN 1 1
.                 (3.7) 
(3.7) describes the dynamics of the behavior of B  in the 
12 +r
R  under the assumption that 
)(tI i  is constant (otherwise the ‘force field’ would not have been conserved and only 
space dependent potential expressions could not have been brought in the dynamics). 
Also the time scale is very small. φ  as given by (3.5) will have to be minimized 
(minimization of energy is an important criterion for neural computation (Laughlin 
2004)), which means 
     )}12(,.....,1{,0
.                                                                          (3.8) 
Combining (3.4) and (3.5) the expression for φ  becomes 
     ∑∑
k ree
))12(),......,1((φφ .                                                                        (3.9) 
If (3.9) is to give a global minimum it should not only be true when φ  is given by (3.9), 
but also it must hold when φ  is given by 
     ∑ ∑
kki reeyx
))12(),......,1((φφ ,                                                                (3.10) 
where )1,1(, ccyx ki +−∈  for some small 0>c . This is because a global minimum is a 
very stable position and therefore under a small perturbation the system always comes 
back to it. Let ikki zyx = . Then combining (3.8) and (3.10) the following is obtained 
     
+++ 0
.....
.....
.....
.....
.....
.....
.....
12,,12,2,112,1,1
2,,2,2,12,1,1
1,,1,2,11,1,1
NmrNmrr
,                                                (3.11) 
where 
))12(),......,1((
. Note that each kiz ,  or ikz  can take 
uncountably infinitely many values from some (small) open interval and for all of them 
(3.11) holds under the perturbation principle. Therefore the mNr ×+ )12(  matrix on the 
left of (3.11) must represent a null linear transformation and it must be a null matrix, 
which implies 
     jki
jki ,,,0
))12(),......,1((
,, ∀=
.                                                      (3.12) 
(3.7) and (3.12) together imply 
     
( ) ( )
= +−=
smi i
.                                                                   (3.13) 
(3.13) says, “Negative and positive experiences in a neuronal network must counter 
balance each other for a stable (which will not be altered due to presence of some amount 
of noise or distraction) unsupervised learning from the experience gained through a new 
behavior.” 
4. Application 
     What do we get from the principle enunciated at the end of the last section in response 
to a strong aversive stimulus? Its mathematical formulation (3.13) says, in the face of 
strong aversive stimuli (such as an imminent danger) the ensuing behavior B  must avoid 
repeating the past behaviors sBi '  with negative experience sM i ' . Note that the stimuli 
can invoke an iM  if and only if iB  is at least partially activated by the stimuli. (3.13) 
says the ensuing behavior B  will have to be such that none of the sBi '  with negative 
experience is repeated. This means in the information space 
12 +r
R  B  will have to sit 
away from each 
iM , which denotes memory of a negative behavior. B  will have its 
own positive and negative parts which will later be stored as new positive and negative 
memories in the network. B  cannot be neutral. This will be the subject of a future work. 
A) Aversion response latency 
     The prefrontal cortex participates in linking perception of stimuli to the guidance of 
behavior including the flexible execution of strategies for obtaining rewards and avoiding 
punishments as an organism interacts with its environment. Recording from neurons 
within healthy tissue in the ventral sites of the right prefrontal cortex short latency (120–
160ms) responses selective for aversive visual stimuli have been observed. No such 
latency was observed for pleasant or neutral stimuli (Kawasaki et al. 2001). (3.13) can 
offer us an explanation of this phenomenon. Aversive stimuli do evoke negative (or 
aversive) memory (that is how the stimuli are identified as aversive, even a novel 
aversive stimulus will have to be decomposable into known aversive features) and 
therefore the ensuing behavior B  must ensure that it has minimum overlap with the 
negative memories in the space 
12 +r
R . This in turn makes sure that B  acts on the 
network as less as possible to repeat the behaviors associated with the negative memories. 
     When there will be only aversive stimuli (no pleasant stimulus) the new behavior will 
have to be organized to avoid the stimuli particularly when the stimuli are strongly 
aversive (like an immediate threat). Clearly a strongly aversive stimulus must invoke the 
memory  
jM  of at least one simple negative behavior with which a strong feeling is to 
be associated. In other words amygdale make sure that the Σ  in (2.11) have larger 
entries. Then there must be the memory 
jM  of a simple positive behavior which can 
take appropriate action in response to the behavior (sensation) corresponding to 
jM . 
Even a new born is hardwired to express displeasure by crying in response to an aversive 
stimulus so that some one else (possibly the mother) is alerted and come in help to avoid 
the stimulus. In order to avoid the aversive stimuli the dynamics of B  will be such that 
(3.13) can take the following form 
     
( ) ( )∑∑ == +
,                                                                          (4.1) 
where )(XF j  is the feeling associated with 
jM  and )(tI j  is the network interaction 
between B  and 
jM , k  is the number of simple positive behavior recalled. This will 
counterbalance the aversive effect of the stimuli and make sure that (3.13) holds. 
     Whereas in case of pleasant or neutral stimuli (without the presence of aversive 
stimuli) avoidance is not necessary, the brain is free to repeat the behaviors with pleasant 
memory. In that case no global minimization of the potential function will be necessary 
and left side of (3.7) does not need to be zero. Therefore equality in (4.1) will also not be 
necessary. The new behavior can sit anywhere and no ‘optimum positioning’ (making the 
potential function globally minimum) will be necessary and this will require less time for 
a response and therefore there will be no latency in the response to pleasant stimuli. 
     When there will be both positive and negative stimuli (3.13) will hold. Thus it appears 
that whenever aversive (negative) stimuli are present either (3.13) or (4.1) needs to be 
calculated. Time taken by this ‘calculation’ in the brain is the reason for latency when 
there are aversive stimuli. Whereas no such latency is necessary for positive or neutral 
stimuli. Probably the brain does not calculate the way shown in this paper. Had it done so 
it would have taken a much longer latency. But the reasoning here is compelling enough 
to conclude that calculations responsible for the observed latency do take place in the 
brain in one form or the other. 
B) Sequential decision making 
     A brain always has to make choices i.e., a behavior is a series of switching or decision 
making procedures. Rabinovich and colleagues have considered the dynamics of decision 
making (DM) by the brain or cognitive state machine (CSM) at the psychological level 
(Rabinovich et al. 2006). Whereas the focus of this paper is on the dynamics of neural 
substrate of such processes. At each DM instant of a CSM the underlying neural 
dynamics of the DM generates the psychological process of DM in the CSM. 
     The dynamics of the CSM is governed by the Lotka-Volterra type equation 
(Rabinovich et al. 2006) 
     )(),( taatIaa i
jijiiii ηρσ +
+−= ∑
& ,                                                             (4.2) 
where )(tai  is the state of the CSM, ),( tIiσ  is a control function which controls the 
dynamics given by (4.3), I  is the input (environmental stimulus), ijρ  is a coupling 
constant between ia  and ja  based on genetic and memorized information (very similar 
to the interaction )(tI i  (2.13) between the memories of the old behavior and the ensuing 
new behavior, which in this paper has been taken to be constant), N  the total number of 
states and )(tiη  is the external noise. Note that in (4.2) the meaning of iσ  and N  are 
different than anywhere else in the earlier part of this paper. 
     
& ,                                                                                                  (4.3) 
where iU  is a potential function and τ  is characteristic time which is very small. 
Difference in states occur at the it  where a decision needs to be made as shown in Figure 
1. ia  can be chosen at an instant jt , where a decision has to be made, as many ways as 
there are minima of iU . 
                             
Figure 1: A sequence of cognitive states. Thin lines are possible paths and the decision 
path has been shown by thick lines. 9643 ,,, tttt  are the instances to make a decision. 
Adopted from (Rabinovich et al. 2006). 
     (3.13) is concerned about the dynamics of neural substrate of each single decision 
making (DM) in the sequence of DM’s shown in Figure 1. (3.13) is also derived by 
minimizing the potential function (3.9). Since the dynamics given by (3.7) is concerned 
about a local decision making in Figure 1, it was possible to take global optimum of the 
potential function given by (3.9) (global optimum of local potential function) and its 
stability ensures that the local dynamics is not perturbed by a small amount of noise. It is 
very important for stable cognition or stable behavior. According to this model the 
alteration in behavior (to make it dynamic) should be introduced by changing feeling 
(given by (2.11)) and synaptic plasticity (given by (2.13)), which happens from state to 
state in a CSM. Therefore despite the global minimum of the local dynamics the CSM 
manages to move on from one state to the next till it reaches the end of life which may be 
signified by lack of emotional impetus. On the other hand at the sequential DM level in a 
CSM (Figure 1) introduction of noise plays a significant role in dynamically altering the 
behavior (equation (4.2)). In both the dynamical systems of this paper and that of 
Rabinovich et al. the time length has been taken to be short. The duration has been taken 
so short that the dynamics of ),( tIiσ  (in (4.3)) and )(tI i  (in (2.13)) do not change 
within that period. In this scenario comparing (3.13) and (4.2) it appears that the external 
noise )(tiη  may take an important role in mediating feeling (emotional distraction or 
attraction with respect to an external stimulus). 
     (3.13) and (4.1) describe how new behavior is formed depending on synaptic 
plasticity, memory and feeling. However there is no deterministic way to compute the 
new behavior from (3.13) or (4.1) even if all the information are available. Also (3.13) or 
(4.1) will be extremely difficult (if not impossible) to compute except for cases like a 
simple behavior (such as inking or gill withdrawal) of a simple animal (such as the sea 
snail Aplysia californica). This is in conformity with the fact that closer is the model to 
the neuronal network level of the brain the more difficult it would be to implement. On 
the other hand the CSM dynamics given by (4.2) is very convenient to compute. 
C) Potential function 
     The potential function introduced by (3.9) in the neuronal network dynamics of the 
brain can be related to metabolic energy consumption during activation of the network to 
execute a behavioral or cognitive task. (3.9) is based on electrophysiology and 
metabolism is related to hemodynamics. Lot of research is going on to find the 
relationship between these two activities of the brain (Logothetis & Wandell 2004). 
Recently a correlation study between stimulus based ERP and fMRI in different parts of 
the human brain has been reported (Gore et al. 2006). Despite convincing evidence of 
their interdependence, such as in the form of dependence of fMRI on ERP or EEG, 
precise knowledge about the nature of dependence is still lacking. A linear transform 
model had been proposed based on the hypothesis that fMRI responses are proportional 
to local average neural activity averaged over a period of time (Boynton et al. 1996). It 
has been reported that in the visual cortex of macaque monkey fMRI response depends 
closely on the local field potential (LFP) (Logothetis et al. 2001). 
     Note that the potential function given by (3.5) can be obtained from (3.7) when the 
time course is short and the quantities on the right side are known. From (3.7) it is clear 
that the potential function φ  (which is defined on the Fourier coefficients of neural spike 
trains (3.9)) depends on past memory, synaptic plasticity and feelings. In case of a 
potential function φ  sum of local components will give the function over the whole 
space, and therefore no matter how large and distributed the network is, local component 
of φ  at a point x  on the cortex (let be denoted by xφ ) can be represented by (3.7) where 
only the neurons in a small neighborhood of x , denoted by )(xn , will participate in the 
computation of xφ . Since xφ  defined on the Fourier coefficients of spike trains of the 
neurons in )(xn  and the value of xφ  depends on past memories, plasticity of synapses 
within )(xn  and part of feeling arising within )(xn  it is supposed to have a role in BOLD 
fMRI response at x  with respect to a stimulus. xφ  may have an affine or linear relation 
with stimulus driven difference in local BOLD signal at x  as modeled in (Logothetis & 
Wandell 2004). But this needs experimentations to establish. 
Conclusion 
     In this paper a system level model of the brain functions for behavior has been 
proposed. The model needs input from individual neurons under the assumption that a 
brain circuit responsible for a behavior can be understood by the behavior of neurons 
alone. This assumption has limitations, for significant change in a neural network may 
occur at the spiking sub-threshold level in the neurons and therefore without changing the 
spike trains (Harris-Warrick & Marder 1991). Note that FFT based information retrieveal 
technique followed in this paper will hold equally good for subthreshold signals, whereas 
spike detection or prediction algorithms may not work for those signals. To account for a 
fuller neural computation synaptic computation (Abbott & Regehr 2004) and dendritic 
computation (London & Hausser 2005) will also have to be incorporated. A close 
investigation into the nature of )(tI  (equation (2.13)) will inevitably call attention to 
synaptic and dendritic computations. The model equation (3.2) will be valid in all 
generality, but the important consequence (3.13) cannot be drawn as easily as shown in 
this paper. 
     The effectiveness of the model dynamics is only for a very short period of time and 
therefore only behaviors in response to strong aversive stimuli have been considered, for 
such response behaviors must have to be very quick. The time duration is so short that it 
has been assumed – the average change in the collection of synapses in a network 
remains fixed within that time. This is an important constraint for introducing potential 
energy function whose global minimization gives stability to the behavioral response in 
the sense that it remains unperturbed in a noisy environment. 
     Even without incorporating synaptic and dendritic computations the model stands 
extremely difficult to implement (for example, calculating iB  and then calculating iM  
will be very challenging). However it can be implemented in case of a simple behavior by 
a simple animal. In invertebrates like the sea snail Aplysia lesser number of neurons will 
have to be monitored (only a few in case of simple behaviors like gill withdrawal or 
inking) and single cell recordings will be less challenging than the mammals. This will 
make the verification of the system possible. In case of human brain functions if the 
behavior of the circuit involved can be monitored to a good extent by recording signals 
only from a few neurons verification of the model may be feasible for some very simple 
behavior like the following. Consider the experiment of tapping the leg (Kandel et al. 
2000). If there is a sharp edge such that dragging the leg too much backward will make it 
bump on the sharp edge which is another aversive stimulus. The final position of the leg 
will be away from the source of tapping as well as the sharp edge. 
     If a relationship between the potential function φ  and the metabolic or hemodynamic 
activity of the brain can be established, study of the system will become easier. 
Introduction of an electrophysiological potential function in the neural computation is a 
significant outcome of the FFT based information extraction method followed in this 
paper. Apart from the prospect of relating it to metabolic energy requirements of the 
brain φ  gives the opportunity of deriving interesting theoretical results as shown in this 
paper. 
     Since the expression of feeling as given by (2.11) is independent of time here the 
detail of feeling did not have to be considered. But feeling is likely to be dependent on 
time in the long run and in that case the entries of Σ  (which is a diagonal matrix) will be 
function of time and the dynamics will be much more complicated. How the eigen values 
of Σ  are controlled by the limbic system of the brain is an open question. No attempt has 
been made in this paper to answer the question. 
Acknowledgement 
     The author thankfully acknowledges the Institute of Mathematical Sciences in 
Chennai, India for a postdoctoral fellowship under which this work has been carried out. 
Some comments by Peter Dayan on a preliminary version of this manuscript are also 
being acknowledged. 
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