402 
MODELING THE OLFACTORY BULB 
COUPLED NONLINEAR OSCILLATORS 
Zhaoping Li J.J. Hopteld* 
 Division of Physics, Mathematics and Astronomy 
*Division of Biology, and Division of Chemistry and Chemical Engineering 
*California Institute of Technology, Pasadena, CA 91125, USA 
*AT&T Bell Laboratories 
ABSTRACT 
The olfactory bulb of mammals aids in the discrimination of 
odors. A mathematical model based on the bulbar anatomy and 
electrophysiology is described. Simulations produce a 35-60 Hz 
modulated activity coherent across the bulb, mimicing the observed 
field potentials. The decision states (for the odor information ) 
here can be thought of as stable cycles, rather than point stable 
states typical of simpler neuro-computing models. Analysis and 
simulations show that a group of coupled non-linear oscillators are 
responsible for the oscillatory activities determined by the odor in- 
put, and that the bulb, with appropriate inputs from higher centers, 
can enhance or suppress the sensitivity to particular odors. The 
model provides a framework in which to understand the transform 
between odor input and the bulbar output to olfactory cortex. 
1. INTRODUCTION 
The olfactory system has a simple cortical intrinsic structure (Shepherd 1979), 
and thus is an ideal candidate to yield insight on the principles of sensory informa- 
tion processing. It includes the receptor cells, the olfactory bulb, and the olfactory 
cortex receiving inputs from the bulb (Figure [1]). Both the bulb and the cortex 
exhibit similar 35-90 Hz rhythmic population activity modulated by breathing. Ef- 
forts have been made to model the bulbar information processing function (Freeman 
1979b, 1979c; Freeman and Schneider 1982; Freeman and Skarda 1985; Baird 1986; 
Skarda and Freeman 1987), which is still unclear (Scott 1986). The bulbar position 
in the olfactory pathway, and the linkage of the oscillatory activity with the sniff 
cycles suggest that the bulb and the oscillation play important roles in the olfactory 
information processing. We will examine how the bulbar oscillation pattern, which 
can be thought of as the decision state about odor information, originates and how 
it depends on the input odor. We then show that with appropriate inputs from the 
higher centers, the bulb can suppress or enhance the its sensitivity to particular 
odors. Much more details of our work are described in other two papers (Li and 
Hopfield 1988a, 1988b). 
Modeling the Olfactory Bulb-Coupled Nonlinear Oscillators 403 
The olfactory bulb has mainly the excitatory mitral and the inhibitory granule 
cells located on different parallel lamina. Odor receptors effectively synapse on the 
mitral cells which interact locally with the granule cells and carry the bulbar outputs 
(Fig 1, Shepherd 1979). A rabbit has about 50,000 mitral, and ~ 10,000,000 granule 
cells (Shepherd 1979). With short odor pulses, the receptor firing rate increases in 
time, and terminates quickly after the odor pulse terminates (Getcheil and Shepherd 
1978). Most inputs from higher brain centers are directed to the granule cells, and 
little is know about them. The surface EEG wave (generated by granule activities, 
Freeman 1978; Freeman and Schneider 1982), depending on odor stimulations and 
animal motivation, shows a high amplitude oscillation arising during the inhalation 
and stopping early in the exhalation. The oscillation is an intrinsic property of the 
bulb itself, and is influenced by central inputs (Freeman 19793; Freeman and Skarda 
1985). It has a peak frequency (which is the same across the bulb) in the range of 
35-90 Hz, and rides on a slow background wave phase locked with the respiratory 
wave. 
2. MODEL ORGANIZATION 
For simplicity, we only include ( N excitatory ) mitral and ( M inhibitory ) 
granule cells in the model. The Receptor input I is -i  -odor,i + -backgroud,i, 
for 1..., N, a superposition of an odor signal -odor and a background input 
-background. -odor  0 increases in time during inhalation, and return expo- 
nentially during exhalation toward the ambient. The central input to the granule 
cells is vector Ic with components Ic, j for I j M. For now, it is assumed that 
Ic ---- 0.1 and Ibackgroud -- 0.243 do not change during a sniff (Li and Hopfield 
1988a). 
Each cell is one unit with its internal state level described by a single vari- 
able, and its output a continuous function of the internal state level. The inter- 
nal states and outputs are respectively X' = {Zl,Z2,...,ZN} and (z(]) -- 
(gz(zi),gz(z2),...,gz(ZN)) (Y-' {]1, ]2,' '' , ]M} and ((Y)- (g(YI), 
g(Y2),..-, g(YM))) for the mitral (granule) cells, where gz _ 0 and g _ 0 are 
the neurons' non-linear sigmoid output functions essential for the bulbar oscillation 
dynamics (Freeman and Skarda 1985) to be studied. 
I odor 
molecules 
olfactory 
bulb 
other 
brain 
olfactory 
cortex 
Receptor inputs 
, ,$ Local 
x :Interaction 
A A A A A A A A 
Central inputs 
Fig. 1. Left: olfactory system; Right: bulbar structure 
Cells marked "+" are mitral cells, "-" are granule cells 
The geometry of bulbar structure is simplified to a one dimensional ring. Each 
cell is specified by an index, e.g. i th mitral cell, and jth granule cell for all i, j 
404 Li and Hopfield 
indicating cell locations on the ring (Fig 1). N x M matrix Ho and M x N 
matrix Wo are used respectively to describe the synaptic strengths (postsynaptic 
input: presynaptic output) from granule cells to mitral cells and vice versa. The 
bulb model system has equations of motion: 
 = -HoG(Y) - zX + I, (2.1) 
' = WoGz(X) - aY + I. 
where ot x - 1/'z, ot - 1/, and 'x - ' -' 7 mzec are the time constants 
of the mitral and granule cells respectively (Freeman and Skarda 1985; Shepherd 
1988). In simulation, weak random noise is added to I .and I to simulate the 
fluctuations in the system. 
3. SIMULATION RESULT 
Computer simulation was done with 10 mitral and granule cells, and show 
that the model can capture the major effects of the real bulb. The rise and fall 
of oscillations with input and the baseline shift wave phase locked with sniff cycles 
are obvious (Fig.2). The simulated EEG (calculated using the approximation by 
Freeman (1980)) and the measured EEG are shown for comparison. During a sniff, 
all the cells oscillate coherently with the same frequency as physiologically observed. 
EEG Wave ]1.0 
Ioom$ 
EEG Wave 
espiratory 
Wave 
Fig.2. A: Simulation result; B: measured result from Freeman and Schneider 1982. 
The model also shows the capability of a pattern classifier. During a sniff, 
some input patterns induce oscillation, while others do not, and different inputs 
induce different oscillation patterns. We showed (Li and Hopfield 1988a) that the 
bulb amplifies the differences between the different inputs to give different output 
patterns, while the responses to same odor inputs with different noise samples differ 
negligibly. 
4. MATHEMATICAL ANALYSIS 
A (damped) oscillator with frequency co can be described by the equations 
:i: = -coy - ox 
or . + 2a5 + (co2 + a2)x = 0 (4.1) 
1 = co z- a y 
Modeling the Olfactory Bulb-Coupled Nonlinear Oscillators 405 
The solution orbit in (z, y) space is a circle if  -- 0 (non-damped oscillator), 
and spirals into the origin otherwise (damped oscillator). If a mitral cell and a 
granule cell are connected to each other, with inputs i(t) and ic(t) respectively, 
then 
k = -h.gy(y) - ax + i(t), 
9 = w. gz(x) - ayy + i(t). (4.2) 
This is the scalar version of equation (2.1) with each upper case letter representing 
a vector or matrix replaced by a lower case letter representing a scalar. It is as- 
sumed that i(t) has a much slower time course than x or y (frequency of sniffs  
characteristic neural oscillation frequency). Use the adiabatic approximation, and 
define the equilibrium point (xo, Yo) as 
ko  0 = -h . gy(Yo) - azXo + i, 
9o  0 = w . g(Xo) - ayyo + ic. 
_ 
Define x I  x Xo,  Y -- Yo. Then 
kl = -h(gy(y) - gy(Yo)) - ax , 
9'= w(g(x) - g(Xo)) - 
(cf. equation (4.1)). If z = y = 0, then the solution orbit 
o+  
/ w(g(s) - g(xo))ds + 
o 
f h(gy(s) - gy(yo))ds = constant 
yo 
is a closed curve in the original (x, y) space surrounding the point (xo, Yo), i.e., 
(x,y) oscillates around the point (Xo,Yo). When the dissipation is included, 
dR/dr < 0, the orbit in (x, y) space will spiral into the point (Xo, Yo). Thus 
a connected pair of mitral and granule celts behaves as a damped non-linear oscilla- 
tor, whose oscillation center (xo, Yo) is determined by the external inputs i and ic. 
For small oscillation amplitudes, it can be approximated by a sinusoidal oscillator 
via linearization around the (xo, Yo): 
k = -h 'gy(Yo)Y - ax (4.4) 
9 = w. g'(Xo)X - y 
where (x, y) is the deviation from (Xo, Yo). The solution is x = roe -atsin(wt+qb) 
= + = + - -- 
ay, which is about right in the bulb, co -- hwgtz(Xo)g(yo). For the bulb, 
a  0.3co. The oscillation frequency depends on the synaptic strengths h and w, 
and is modulated by the receptor and central input via (Xo, Yo). 
406 Li and Hopfield 
N such mitral-granule pairs with cell interconnections between the pairs rep- 
resent a group of N coupled non-linear damped oscillators. This is exactly the 
situation in the olfactory bulb. The locality of synaptic connections in the bulb im- 
plies that the oscillator coupling is also local. (That there are many more granule 
cells than mitral cells only means that there is more than one granule cell in each 
oscillator.) Corresponding to equation (4.2) and (4.4), we have equation (2.1) and 
= - -- H - 
 = WoG(Xo)X- ayY =- WX - ayY. 
where (X, Y) are now deviations from (Xo, Yo) and Gtz(Xo) 
diagonal matrices with elements: [Gtz(Xo)]ii = gtz(Xi,o) _> 
gy(Yj,o) >_ O, for all i,j. Eliminating Y, 
and G(Yo) are 
o, [G(Y0)lji = 
J( + (a + a)J( + (A + aa)X = 0 
(4.6) 
where A --= HW ' ' . i th 
= HoCy(Yo)WoCz(Xo) The oscillator (mitral cell) follows 
the equation 
 i + (otz + oql)5i +(Aii + otzoty)xi +  Aijxj = 0 
(cf. equation (4.1)), the the last term describes the coupling between oscillators. 
Non-linear effect occurs when the amplitude is large, and make the oscillation wave 
form non-sinusoidal. 
If X k is one of the eigenvectors of A with eigenvalue k, equation (4.6) has 
k th oscillation mode 
X oc Xe i""t = Xexp(- 
(az + au) t 4- iA/ + (az - au)2 t) (4.8) 
2 4 
Components of X k indicate oscillators' relative amplitudes and phases (for each 
k -- 1,2,...,N independent mode). For simplicity, we set 
then X cr Xk e-at+iv/J. Each mode has frequency ReVk, where Re means 
the real part of a complex number. If Se(-c -t- iV/k) > 0 is satisfied for 
some k, then the amplitude of the k th mode will increase with time, i.e. growing 
oscillation. Starting from an initial condition of arbitrary small amplitudes in linear 
analysis, the mode with the fastest growing amplitude will dominate the output, 
and the whole bulb will oscillate in the same frequency as observed physiologically 
(Freeman 1978; Freeman and Schneider 1982) as well as in the simulation. With the 
non-linear effect, the strongest mode will suppress the others, and the final activity 
output will be a single "mode" in a non-linear regime. 
Modeling the Olfactory Bulb Coupled Nonlinear Oscillators 407 
Because of the coupling between the (damped) oscillators, the equilibrium point 
(Xo, Yo) of a group of oscillators is no longer always stable with the possibility 
of growing oscillation modes. 'k must be complex in order to have k th mode 
grow. For this, a necessary (but not sufficient) condition is that matrix A is non- 
symmetric. Those inputs that make matrix A less symmetric will more likely induce 
the oscillatory output and thus presumpably be noticed by the following olfactory 
cortex (see Li and Hopfield 1988a for details). 
The consequences (also observed physiologically) of our model are (Freeman 
1975,1978; Freeman and Schneider 1982; Li and Hopfield 1988a): 1): local mitral 
cells' oscillation phase leads that of the local granule cells by a quarter cycle; 2): 
oscillations across the bulb have the same dominant frequency whose range possible 
should be narrow; 3): there should be a non-zero phase gradient field across the 
bulb; 4): the oscillation activity will rise during the inhale and fall at exhale, and 
rides on a slow background baseline shift wave phase locked with the sniff cycles. 
This model of the olfactory bulb can be generalized to other masses of inter- 
acting excitatory and inhibitory cells such as those in olfactory cortex, neocortex 
and hippocampus (Shepherd 1979) etc. where there may be connections between 
the excitatory cells as well as the inhibitory cells (Li and Hopfield 1988a). Suppose 
that Bo and '0 are excitatory-to-excitatory and inhibitory-to-inhibitory connec- 
tion matrices respectively, then equation (4.6) becomes 
.' + (az - B + ay + C)J + (A + (az - B)(ay + C))X = 0 (4.9) 
where B = BoGz(Xo) and C  HCoG(Yo)H -. 
5. COMPUTATIONS IN THE OLFACTORY BULB 
Receptor input I influences (Xo, Yo) as follows 
dXo -. (a  + HW)-X(adI + d) 
dYo  (a  + WH)-X(WdI- aH-Xd) 
This is how the odor input determines the bulbar output. Increasing -[odor not only 
raises the mean activity level (Xo, Yo) (and thus the gain (Gz(Xo), a(Yo))), but 
also slowly changes the oscillation modes by structurally changing the oscillation 
equation (4.6) through matrix A = Ho(Yo)Woz(Xo). If (Xo, Yo) is raised 
to such an extent that Re(-a+ iv) > 0 is satisfied for some mode k, the equi- 
librium point (Xo, Yo) becomes unstable and this mode emerges with oscillatory 
bursts. Different oscillation modes that emerge are indicative of the different odor 
inputs controlling the system parameters (Xo, Yo), and can be thought of as the 
decision states reached for odor information, i.e., the oscillation pattern classifies 
odors. When (Xo, Yo) is very low (e.g. before inhale), all modes are damped, and 
only small amplitude oscillations occur, driven by noise and the weak time variation 
of the odor input. The absence of oscillation can be interpreted by higher processing 
408 Li and Hopfield 
centers as the absence of an odor (Skarda and Freeman 1987). Detailed analysis 
shows how the bulb selectively responds (or not to respond) to certain input pat- 
terns (Li and Hopfield 1988a) by choosing the synaptic connections appropriately. 
This means the bulb can have non-uniform sensitivities to different odor receptor 
inputs and achieve better odor discriminations. 
6. PERFORMANCE OPTIMIZATION IN THE BULB 
We discussed (Li and Hopfield 1988a) how the olfactory bulb makes the least 
information contamination between sniffs and changes the motivation level for odor 
discrimination. We further postulate with our model that the bulb, with appropriate 
inputs from the higher centers, can enhance or suppress the sensitivity to particular 
odors (details in Li and Hopfield 1988b). When the central input Ic is not fixed, it 
can control the bulbar output by shifting (Xo, Yo), just as the odor input I can, 
equation (5.1) becomes: 
dXo . (a  + HW)-X(adI + dJ  - HdI + aW-Xdr) 
dYo -, (a  + WH)-X(WdI - aH-XdJ  + adI + dr) 
Suppose that I  I,backgroun d + I,ontro I where I,ontro I is the control sig- 
nal which changes during a sniff. Olfactory adaptation is achieved by having an 
Ic,coatro I = Ic ca'cel which cancels the effect of Iodor on Xo --cancelling. This 
keeps the mitral cells baseline output (z(Xo) and gain ((Xo) low, and thus 
makes the oscillation output impossible as if no odor exists. We can then expect 
that reversing the sign of Ic caacel will cause the bulb to have an enhanced, instead 
of reduced (adapted), response to Iodor m anti-cancelling, and achieve the olfac- 
tory enhancement. We can derive further phenomena such as recognizing an odor 
component in an odor mixture, cross-adaptation and cross-enhancement (Li and 
Hopfield 1988b). Computer simulations confirmed the expected results. 
7. DISCUSSION 
Our model of the olfactory bulb is a simplification of the known anatomy 
and physiology. The net of the mitral and granule cells simulates a group of cou- 
pled non-linear oscillators which are the sources of the rhythmic activities in the 
bulb. The coupling makes the oscillation coherent across the bulb surface for each 
sniff. The model suggests, in agreement with Freeman and coworkers, that stability 
change bifurcation is used for the bulbar oscillator system to decide primitively on 
the relevance of the receptor input information. Different non-damping oscillation 
modes emerged are used to distinguish the different odor input information which 
is the driving source for the bifurcations, and are approximately thought of as the 
(unitary) decision states of the system for the odor information. With the extra 
information represented in the oscillation phases of the cells, the bulb emphasizes 
the differences between different input patterns (section 4). Both the analysis and 
simulation show that the bulb is selectively sensitive to different receptor input pat- 
terns. This selectivity as well as the motivation level of the animal could also be 
Modeling the Olfactory Bulb-Coupled Nonlinear Oscillators 409 
modulated from higher centers. This model also successfully applies to bulbar abil- 
ity to use input from higher centers to suppress or enhance sensitivity to particular 
target or to mask odors. 
This model does not exclude the possibility that the information be coded in 
the non-oscillatory slow wave Xo which is also determined by the odor input. The 
chief behaviors do not depend on the number of cells in the model. The model can 
be generalized to olfactory cortex, hippocampus and neocortex etc. where there are 
more varieties of synaptic organizations. 
Acknowledgements 
This research was supported by ONR contract N00014-87-K-0377. We would also 
like to acknowledge discussions with J.A. Bower. 
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