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Nonlinear Simulations of Olfactory System Mohammed Karim Natacha Gueorguieva Iren Valova*
Abstract
1. Introduction An important part of human anatomy is the olfactory bulb in the front-lower part of the brain, which is responsible for processing the smell signals. The bulb does the most processing in the olfactory pathway of smell signals, which are then send to upper parts of the brain for further processing and perhaps responding to the signal. There is much interest in the olfactory system for scientists because most of the sense systems (i.e. sight, hearing) function the same way but are more complex. The olfactory system, especially the bulb, is phylogenetically primitive. Not only that, the bulb has been found in most other animals with the responsibility of primary functions aside from smell processing. The mouse, for example, has a much similar bulb as a human, but it’s much bigger in size and performs many functions. Most animals have a strong sense of smell through which they generally search for food, which is basically due to the bulbar activity. In scientific advancements, it is essential that sufficient experiments be conducted to conclude the best solution from all the possible outcomes. Biological development relating to human structure is one of the primary fields that can only accept the best solution. That is simply because biological experiments cannot be fluently conducted on human beings. The person must have died recently (for the most part), who formerly agreed to be used for scientific development, and the experiments can be done only limited number of times because the dissection of the body can only be done once, after which the structure no longer behaves the same. The human brain is the most critical of all the parts. It is located in a hard to reach place, guarded from all sides by the scull and layers of essential tissues. Most importantly its activity is based on rapid electrical impulses, which are extremely difficult to implement on an open brain [5] [ 6] [7]. The activity in the bulb is oscillatory and so the model is based on implementing these behaviors with equations of coupled oscillators and in correlation to the actual cell structures, their behavior and their connections.
2. The Model The purpose of the model is to simulate the olfactory processing with mathematical representation, which would enable its analysis and classification. The model attempts to simulate the three major levels of the olfactory pathway: the bulb, the anterior nucleus, and the prepyriform cortex. Each of these sections is simulated as separate neural systems with interactions between them [8]. The computer model uses coupled oscillators because the cell structures act like coupled oscillators. These coupled oscillators however, are numerous in number. There are thousands of mitral and granule cells in the bulb. In order to get an understanding of the system at large, the model must simulate the coupling of some number of these cells. In order to implement this in the model, we use nonlinear differential equations with matrices to represent a system of mitral and granule cells [9] [10]. We will use coupled ten mitral cells and ten granule cells. The governing set of equations for the bulb system is:
The X and Y are vectors with ten elements representing
the impulse, or charge, state of mitral and granule cells respectively. Each mitral cell
is coupled with a neighboring granule cell and the change in its state will depend on the
neighboring cells’ output activity and their connection strength, and vice versa
(granule to mitral). The functions This system of equations incorporates most of the behavior for the bulb. The input from the central brain is kept constant because the model analyzes the bulb activity as standalone and the input is assumed to be insignificant at this stage. The output to the central brain does not change the bulbar activity, so it is not implemented. The output function of each cell (g) is a simple sigmoid function. Any equation representing nonlinear sigmoid function is sufficient to simulate the output of each cell with respect to its internal state. The fact that this is a nonlinear function is essential for oscillation of the system. The equation used in our model is:
Both
Excitatory [ a (output factor) = 1.55
b (curve constant) = 4
c (x shift) = -2
d (y shift) = 0
Inhibitory [ a (output factor) = 3.15 b (curve constant) = 2 c (x shift) = -3 d (y shift) = 0
The matrices Ho and Wo are the coupling strengths between the excitatory and inhibitory cells. Following are the values assigned to them:
These are almost diagonal matrices representing the neighboring connections in the bulb, and it wraps around at the corners, thus simulating a ring of coupled oscillators. This ring is the basis for understanding the simpler model at first, and it may not seem related to the structure of the bulb, but it does correspond. The actual bulb has thousands of cells and if analyzing any number of them around the center of the bulb, they receive oscillation stimulations from many sides, which is similar to the behavior of the oscillating ring. The only deficiency is that the ring does not fully implement the lateral two-dimensional connections, which supposedly however, does not change the behavior significantly. The All these parameters in the system equations are used to simulate the
oscillatory activity of the olfactory bulb. The other two sections (layers) are also
simulated in the similar way. All the constants are the same ( Each section of the olfactory pathway is represented by the pair of
equations (1 and 2). The inputs The parameters into this model were assigned values from the Freeman’s model [3] [4]. The implementation of the multiple levels with same oscillatory mechanism, required adjustment for desired behavior. The functions of interaction between the levels are crucial for correct simulation, and the use of output (g) functions gave good results.
3. Simulations The program is written for the Matlab application, using the various matrix utilities and the plotting functionality. The model's sets of governing equations are solved using the simple Euler Method. The olfactory simulation results are dependent on the input odor, which is exclusively selected from the four odors in the beginning of the program by un-commenting it. The simulation lasts one sniff cycle: 450 milliseconds. The differentiation of the governing equations with respect to time is taken per every "time_step" interval, which is set in the beginning of the program as well, and changing the time_step can result in variations of accuracy. These are the most volatile settings in the program (namely the odor and time_step), which are varied to control the simulation. Lastly, the simulation is conducted over the 450 milliseconds of time span, and the result of all the cells in each layer is stored in matrices for each moment in time, which is then plotted with respect to time.
Fig.2. Freeman olfactory bulb model
The input odor is a ten-element vector that represents the inputs that will be processed with noise and supplied as sensory input to the ten mitral cells in the bulb. Central brain input is initialized to zero for all time, and noise from
the central brain into the anterior layer is given random values. The input from the
second layer to the first layer ( A fraction of this current output of cells is used to compute the
interaction inputs ( The displaying of results is done first for the input information. About twenty figures are plotted and each one is given axes labels and titles. The odor that was inputted in the simulation is displayed as a bar graph. A sample input supplied to the first Mitral cell in the first layer is also plotted for the time span after the input processing is complete, and the input to all the mitral cells is similarly plotted. For each successive layer, the activation of each cell is plotted in a sub-graph, and then a three-dimensional graph relating all the cells is plotted. Also the calculated EEG is plotted.
4. Analysis of Simulation Results The model gives satisfying results in simulating the bulb. The behavior of the system of cells demonstrates well-ordered oscillations and the affect of odor. Fig. 3 shows the activations of all the ten mitral cells over time, individually and collectively.
The activation of all the cells is different and not only is it different, it is harmonious. The three-dimensional graph (Fig. 4) shows that the oscillations are coherent and corresponding to the sniff cycle. The following graphs (Fig. 5) show the activations of the granule cells over the time:
The model of the olfactory bulb is not exactly the same as the actual behavior of the bulb, but it sufficiently demonstrates its behavior close enough to make an elaborate model of the whole system feasible.
5. Conclusions It can be observed that the olfactory bulb responds directly with the input odor, and the anterior nucleus being the mediator between the bulb and other communications, is somewhat responsive to the odor, and prepyriform cortex is comparatively the least responsive to the odor. That is not to say that it’s response is not strong or clear enough, but rather that its activity does not directly resemble the input odor. The activation of the excitatory cells is the most concerning because these cells send the output synapses signal to upper parts of the brain. This can be very important in understanding the brain functionality. The activation of the prepyriform cortex, being somewhat independent as it is, can be used to classify the odor, that is: is it toxic, is it flammable, is it sweet perfume, etc. The model is able to reduce noise, allowing the pattern to emerge from incomplete and noisy input. The simulations show that it is capable to solve the odor detection, recognition and segmentation problems [11]. The output of the bulb and the anterior nucleus sends the actual sensed odor to the upper parts of the brain for further processing, according to the activation of the prepyriform cortex. All these hypotheses are exciting and may give us a comprehensive understanding of the brain. They, however, require extensive work to verify their validity.
6. Acknowledgments This work is funded in part by PSC-CUNY Awards #61782-00-30, #63374-00-32 from City University of New York, and by UMD Foundation grant #UMDF-525360, University of Massachusetts Dartmouth.
References [1] Chang Hung-Jen, Freeman W. J., and Burke B., Local Homeostasis stabilizes a model of the olfactory system globally in respect to perturbations by input during pattern classification. Intern. Journal of Bifurcation and Chaos, Vol. 8, No 11, 2107-2123, 1998. [2] Freeman W. J., Nonlinear gain mediating cortical stimulus-response relations. Biol. Cybernetics 33, 237-247, 1979. [3] Freeman W. J., Simulation of chaotic EEG patterns with a dynamic model of the olfactory system. Biol. Cybernetics 56, 139-150, 1987 [4] Freeman W. J., Random activity at the microscopic neural level in cortex ("noise") sustains and is regulated by low-dimensional dynamics of microscopic cortical activity ("chaos"), Int. J. Neural Systems 7, 473-480, 1996. [5] Gordon Shepherd, The synaptic organization of the brain. Oxford, ISBN 0-19-511824-3, 2001. [6] Li Z., and Hopfield J.J., Modeling the olfactory bulb and its neural oscillatory processing. Biol. Cybernetics 61, 379-392, 1989. [7] Netter F. H., Atlas of Human Anatomy, Novartis, ISBN 0914168193, 1989. [8] Osamu Hoshino, Y. Kashimori, T. Kambara, An olfactory recognition model based on spatio-temporal encoding of odor quality in the olfactory bulb. Biol. Cybernetics, 79, 109-120, 1998. [9] Juergen Pahle, Iren Valova, George Georgiev, and Natacha Georgieva, Oscillatory Simulation of Mitral and Granule Cell Behavior in the Olfactory Bulb. Proceedings of International Conference on Artificial Intelligence (IC-AI'2000), Las Vegas, Nevada, June 26 - 29, 2000, pp. 483-489. [10] Zhaoping Li and John Hertz, Odor recognition and segmentation by a model olfactory bulb and cortex. Computation in Neural Systems, 11, p. 83-102, 2000. [11] Valova, N. Georgieva, Y. Kosugi, Modeling of the Odor Information Processing in the Mammalian Brain, Proceedings of the International Joint Conference on Neural Networks (IJCNN’2001), Washington DC, USA, CD-ROM - ID# 257, 2001. |
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ASEE 
(1)
(2)
and 
are
vectors whose elements represent the output of each cell according to its state,
calculated by functions 
and 
for each element. The two matrices 
and 
represent the coupling strengths. The change in each cell’s state
involves the addition of impulses that are sent to it from coupled cells. The negative
sign in front of 
and 
represent the decay factor of each
cell, and thus the change in its state would decrease in proportion of decay; 
in the mitral equation is the primary input of the receptor cell’s
fibers from the odor, and 
is the input to granule cells from the
other levels of the olfactory pathway and the central brain. 
Fig. 1. Excitatory
and inhibitory graphs
and 
are simply the reciprocal of the time constant: the time it takes for
each cell to lose its charge. This is the same for mitral and granule cells: 7 ms; thus 
and 
are 1/7 since the
differentiation is done in the unit of milliseconds. 
, 
)
are also the same. The objective is to simulate the other sections with similar coupled
oscillators as in the bulb. In the actual biological system, all sections apparently have
homogeneous neural activity with many variations. We assume them to be the same for
computational simplicity, and we intend to account for the variations by adjusting the
interactions between them according to the differences. 
and 
, however, are different for each section, and they represent the
interactions between them. Only the mitral cells of the bulb receive input from the
receptor cells, so 
), which is the activation of anterior cells with influence from the
output of prepyriform, and it is added with input from deep central brain, which is
assumed to be zero. The 
from the first level (first section, the bulb).
The input 
), which is 
for each moment in time. The inhibitory cells' states are stored as a
matrix called 
similarly over the time. For the first, second, and
third layers they are named 
, 
, and 
respectively. And similarly the
variables to store the change in each cell over one time interval are called 
and 
, 
and 
, etc. These are vectors of ten elements; they need not store values
over the whole time. Finally, the values for the variables of each layer at the first time
instance are initialized to random noise [9] [11]. 
and 
. Finally, the change added with the cell's current activation is
stored in the cell's next time interval. This change is multiplied with time_step to give
appropriate differentiation. This procedure occurs repeatedly in the loop for the number
of time intervals. When the loop finishes, the result of the activations is stored in
matrices of each layer for the time span. 
