Simulation of a Thalamocortical Circuit for 
Computing Directional Heading in the Rat 
Hugh T. Blair* 
Department of Psychology 
Yale University 
New Haven, CT 06520-8205 
tadb @minerva. cis.yale.edu 
Abstract 
Several regions of the rat brain contain neurons known as head-direc- 
tion cells, which encode the animal's directional heading during spatial 
navigation. This paper presents a biophysical model of head-direction 
cell activity, which suggests that a thalamocortical circuit might com- 
pute the rat's head direction by integrating the angular velocity of the 
head over time. The model was implemented using the neural simulator 
NEURON, and makes testable predictions about the structure and func- 
tion of the rat head-direction circuit. 
1 HEAD-DIRECTION CELLS 
As a rat navigates through space, neurons called head-direction cells encode the animai's 
directional heading in the horizontal plane (Ranck, 1984; Taube, Muller, & Ranck, 1990). 
Head-direction cells have been recorded in several brain areas, including the postsubicu- 
lum (Ranck, 1984) and anterior thalamus (Taube, 1995). A variety of theories have pro- 
posed that head-direction cells might play an important role in spatial learning and 
navigation (Brown & Sharp, 1995; Burgess, Recce, & O'Keefe, 1994; McNaughton, 
Knierim, & Wilson, 1995; Wan, Touretzky, & Redish, 1994; Zhang, 1995). 
1.1 BASIC FIRING PROPERTIES 
A head-direction cell fires action potentials only when the rat's head is facing in a particu- 
lar direction with respect to the static surrounding environment, regardless of the animal's 
location within that environment. Head-direction cells are not influenced by the position 
of the rat's head with respect to its body, they are only influenced by the direction of the 
*Also at the Yale Neuroengineering and Neuroscience Center (NNC), 5 Science Park North, New 
Haven, CT 06511 
Simulation of Thalamocortical Circuit for Computing Directional Heading in Rats 153 
10 
'c 05 
x 
360,0 
270 90 
90 80 270 360 
Head Drecton 180 
Figure 1' Directional Tuning Curve of a Head-Direction Cell 
head with respect to the stationary reference frame of the spatial environment. Each head- 
direction cell has its own directional preference, so that together, the entire population of 
cells can encode any direction that the animal is facing. 
Figure I shows an example of a head-direction cell's directional tuning curve, which plots 
the firing rate of the cell as a function of the rat's momentary head direction. The tuning 
curve shows that this cell fires maximally when the rat's head is facing in a preferred 
direction of about 160 degrees. The cell fires less rapidly for directions close to 160 
degrees, and stops firing altogether for directions that are far from 160 degrees. 
1.2 THE VELOCITY INTEGRATION HYPOTHESIS 
McNaughton, Chen, & Markus (1991) have proposed that head-direction cells might rely 
on a process of dead-reckoning to calculate the rat's current head direction, based on the 
previous head direction and the angular velocity at which the head is turning. That is, 
head-direction cells might compute the directional position of the head by integrating the 
angular velocity of the head over time. This velocity integration hypothesis is supported 
by three experimental findings. First, several brain regions that are associated with head- 
direction cells contain angular velocity cells, neurons that fire in proportion to the angular 
head velocity (McNaughton et al., 1994; Sharp, in press). Second, some head-direction 
cells in postsubiculum are modulated by angular head velocity, such that their peak firing 
rate is higher if the head is turning in one direction than in the other (Taube et al., 1990). 
Third, it has recently been found that head-direction cells in the anterior thalamus, but not 
the postsubiculum, anticipate the future direction of the rat's head (Blair & Sharp, 1995). 
1.3 ANTICIPATORY HEAD-DIRECTION CELLS 
Blair and Sharp (1995) discovered that head-direction cells in the anterior thalamus shift 
their directional preference to the left during clockwise turns, and to the right during coun- 
terclockwise turns. They showed that this shift occurs systematically as a function of head 
velocity, in a way that allows these cells anticipate the future direction of the rat's head. 
To illustrate this, consider a cell that fires whenever the head will be facing a specific 
direction, 0, in the near future. How would such a cell behave? There are three cases to 
consider. First, imagine that the rat's head is turning clockwise, approaching the direction 
0 from the left side. In this case, the anticipatory cell must fire when the head is facing to 
the left of 0, because being to the left of 0 and turning clockwise predicts arrival at 0 in the 
near future. Second, when the head is turning counterclockwise and approaching 0 from 
the right side, the anticipatory cell must fire when the head is to the right of 0. Third, if the 
head is still, then the cell should only fire if the head is presently facing 0. 
In summary, an anticipatory head direction cell should shift its directional preference to 
the left during clockwise turns, to the right during counterclockwise turns, and not at all 
when the head is still. This behavior can be formalized by the equation 
Ix(v) = 0 - v'r, [ 1] 
154 H.T. BLAIR 
where g denotes the cell's preferred present head direction, v denotes the angular velocity 
of the head, 0 denotes the future head direction that the cell anticipates, and 'c is a constant 
time delay by which the cell's activity anticipates arrival at 0. Equation I assumes that g 
is measured in degrees. which increase in the clockwise direction, and that v is positive for 
clockwise head turns. and negative for counterclockwise head tums. Blair & Sharp (1995) 
have demonstrated that Equation I provides a good approximation of head-direction cell 
behavior in the anterior thalamus. 
1.3 ANTICIPATORY TIME DELAY () 
Initial reports suggested that head-direction cells in the anterior thalamus anticipate the 
future head direction by an average time delay of : = 40 msec, whereas postsubicular cells 
encode the present head direction, and therefore "anticipate" by : = 0 msec (Blair & 
Sharp, 1995; Taube & Muller, 1995). However, recent evidence suggests that individual 
neurons in the anterior thalamus may be temporally tuned to anticipate the rat's future 
head-direction by different time delays between 0-100 msec, and that postsubicular cells 
may "lag behind" the present head-direction by about 10 msec (Blair & Sharp, 1996). 
2 A BIOPHYSICAL MODEL 
This section describes a biophysical model that accounts for the properties of head-direc- 
tion cells in postsubiculum and anterior thalamus, by proposing that they might be con- 
nected to form a thalamocortical circuit. The next section presents simulation results from 
an implementation of the model, using the neural simulator NEURON (Hines, 1993). 
2.1 NEURAL ELEMENTS 
Figure 2 illustrates a basic circuit for computing the rat's head-direction. The circuit con- 
sists of five types of cells: 1) Present Head-Direction (PHD) Cells encode the present 
direction of the rat's head, 2) Anticipator), Head-Direction (AHD) Cells encode the future 
direction of the rat's head, 3)Angular-Velocity (AV) Cells encode the angular velocity of 
the rat's head (the CLK AV Cell is active during clockwise turns, and the CNT AV Cell is 
active during counterclockwise turns), 4) the Angular Speed (AS) Cell fires in inverse pro- 
portion to the angular speed of the head, regardless of the turning direction (that is, the AS 
Cell rites at a lower rate during fast tums, and at a higher rate during slow turns), 5) Angu- 
lar-Velocity Modulated Head-Direction (AVHD) Cells are head-direction cells that fire 
' I 
AT = Antedor Thalamus 
UB = Uammillary Bodies 
PS = Postsubiculum 
 Excitatory ---e Inhibitory RS = Reb'osplenial Cortex 
RTN = Reticular Thai. Nu. 
Figure 2: A Model of the Rat Head-Direction System 
Simulation of Thalamocortical Circuit for Computing Directional Heading in Rats 155 
only when the head is turning in one direction and not the other (the CLK AVHD Cell fires 
in its preferred direction only when the head is turning clockwise, and the CNT AVHD 
Cell fires in its preferred direction only when the head turns counterclockwise). 
2.2 FUNCTIONAL CHARACTERISTICS 
In the model, AHD Cells directly excite their neighbors on either side, but indirectly 
inhibit these same neighbors via the AVHD Cells, which act as inhibitory interneurons. 
AHD Cells also send excitatory feedback connections to themselves (omitted from Figure 
2 for clarity), so that once they become active, they remain active until they are turned off 
by inhibitory input (the rate of firing can also be modulated by inhibitory input). When 
the rat is not turning its head, the cell representing the current head direction rites con- 
stantly, both exciting and inhibiting its neighbors. In the steady-state condition (i.e., when 
the rat is not turning its head), lateral inhibition exceeds lateral excitation, and therefore 
activity does not spread in either direction through the layer of AHD Cells. However, 
when the rat begins turning its head, some of the AVHD Cells are turned off, allowing 
activity to spread in one direction. For example, during a clockwise head turn, the CLK 
AV Cell becomes active, and inhibits the layer of CNT AVHD Cells. As a result, AHD 
Cells stop inhibiting their right neighbors, so activity spreads to the right through the layer 
of AHD Cells. Because AHD Cells continue to inhibit their neighbors to the left, activity 
is shut down in the leftward direction, in the wake of the activity spreading to the right. 
The speed of propagation through the AHD layer is governed by the AS Cell. During 
slow head turns, the AS Cell fires at a high rate, strongly inhibiting the AHD Cells, and 
thereby slowing the speed of propagation. During fast head turns, the AS Cell rites at a 
low rate, weakly inhibiting the AHD Cells, allowing activity to propagate more quickly. 
Because of inhibition from AS cells, AHD cells fire faster when the head is turning than 
when it is still (see Figure 4), in agreement with experimental data (Blair & Sharp, 1995). 
AHD Cells send a topographic projection to PHD Cells, such that each PHD Cell receives 
excitatory input from an AHD Cell that anticipates when the head will soon be facing in 
the PHD Cell's preferred direction. AHD Cell activity anticipates PHD Cell activity 
because there is a transmission delay between the AHD and PHD Cells (assumed to be 5 
msec in the simulations presented below). Also, the weights of the connections from 
AHD Cells to PHD Cells are small, so each AHD Cell must fire several action potentials 
before its targeted PHD Cell can begin to fire. The time delay between AHD and PHD 
Cells accounts for anticipatory firing, and corresponds to the ' parameter in Equation 1. 
2.3 ANATOMICAL CHARACTERISTICS 
Each component of the model is assumed to reside in a specific brain region. AHD and 
PHD Cells are assumed to reside in anterior thalamus (AT) and postsubiculum (PS), 
respectively. AS Cells have been observed in PS (Sharp, in press) and retrospleniai cortex 
(RS) (McNaughton, Green, & Mizumori, 1986), but the model predicts that they may also 
be found in the mammiilary bodies (MB), since MB receives input from PS and RS (Shi- 
bata, 1989), and MB projects to ATN. AVHD Cells have been observed in PS (Taube et 
al., 1990), but the model predicts that they may also be found in the reticular thalamic 
nucleus (RTN), because RTN receives input from PS/RS (Lozsadi, 1994), and RTN inhib- 
its AT. It should be noted that lateral excitation between ATN cells has not been shown, so 
this feature of the model may be incorrect. Table I summarizes anatomical evidence. 
3 SIMULATION RESULTS 
The model illustrated in Figure 2 has been implemented using the neural simulator NEU- 
RON (Hines, 1993). Each neural element was represented as a single spherical cornpart- 
156 H.T. BLAIR 
Table l' Anatomical Features of the Model 
FEATURE OF MODEL 
PHD Cells in PS/RS 
AHD Cells in AT 
AV Cells in PS/RS 
AT projects to PS 
AT projects to RTN 
PS/RS projects to RTN 
AVHD Cells in RTN 
AS Cells in MB 
[rEFERENCE 
Chen et al., 1990; Ranck, 1984 
Blair & Sharp, 1995 
McNaughton et al., 1994; Sharp, in press 
van Groen & Wyss, 1990 
Shibata, 1992 
Lozsadi, 1994 
PREDICTION OF MODEL 
PREDICTION OF MODEL 
ment, 30 gm in diameter, with RC time constants ranging between 15 and 30 msec. Syn- 
aptic connections were simulated using triggered alpha-function conductances. The 
results presented here demonstrate the behavior of the model, and compare the properties 
of the model with experimental data. 
To begin each simulation, a small current was injected in to one of the AHD Cells, causing 
it to initiate sustained firing. This cell represented the simulated rat's initial head direc- 
tion. Head-turning behavior was simulated by injecting current into the AV and AS Cells, 
with an amplitude that yielded firing proportional to the desired angular head velocity. 
3.1 ACTIVITY OF HEAD-DIRECTION CELLS 
Figure 3 presents a simple simulation, which illustrates the behavior of head-direction 
cells in the model. The simulated rat begins by facing in the direction of 0 degrees. Over 
the course of 250 msec, the rat quickly turns its head 60 degrees to the right, and then 
returns to the initial starting position of 0 degrees. The average velocity of the head in this 
simulation was 480 degrees/sec, which is similar to the speed at which an actual rat per- 
forms a fast head turn (Blair & Sharp, 1995). Over the course of the simulation, neural 
activation propagates from the O-degree cell to the 60-degree cell, and then back to the 0- 
degree cell. 
3.2 COMPARISON WITH EXPERIMENTAL DATA 
To examine how well the model reproduces firing properties of PS and AT cells, another 
simple simulation was performed. The firing rate the model's PHD and AHD Cells was 
examined while the simulated rat performed several 360-degree revolutions in both the 
clockwise and counterclockwise directions. Results are summarized in Figure 4, which 
ACTIVITY OF PHD CELLS ANIMAL 
151 IIIIII 
48' 
Cell 
6O  
Cell 
BEHAVIOR 
o' 
Turning Right Turning Lelt 
Time (msec) 
Figure 3: Simulation Example 
Average Angular Velocity = 480"/aec 
Simulation of Thalamocortical Circuit for Computing Directional Heading in Rats 15 7 
O-OAT (exper.lmal lta} 
C) -i'l PS (exp4imlntld G&III) -:) 
 -' AT (mol Ol ..-'" 
H PS (1 t ....'" 
1  2 3 4 5 
Angular Head Veli (desec) 
25.0 , 
20.0  
 0 o 
15.0  
10.0  -- 
5.0 i 0 ...................... 
0.0 
0 1  200 300 4 500 
Angular Head Veloci (ds) 
Figure 4: Compared Properties of Real and Simulated Head-Direction Cells 
compares simulation data with experimental data. The experimental data in Figure 4 
shows averaged results for 21 cells recorded in AT, and 19 cells recorded in PS. 
Because AT cells anticipate the future head direction, they exhibit an angular separation 
between their clockwise and counterclockwise directional preference, whereas as no such 
separation occurs for PS cells (see section 2.4). For AT cells, the magnitude of the angular 
separation is proportional to angular head velocity, with greater separation occurring for 
fast tums, and less separation for slow tums (see Eq. 1). The left panel of Figure 4 shows 
that the model's PHD and AHD Cells exhibit a similar pattern of angular separation. 
Blair & Sharp (1995) reported that the firing rates of AT and PS cells differ in two ways: 
1) AT cells fire at a higher rate than PS cells, and 2) AT cells have a higher rate during fast 
tums than during slow tums, whereas PS cells fire at the same rate, regardless of turning 
speed. In Figure 4 (right panel), it can be seen that the model reproduces these findings. 
4 DISCUSSION AND CONCLUSIONS 
In this paper, I have presented a neural model of the rat head-direction system. The model 
includes neural elements whose firing properties are similar to those of actual neurons in 
the rat brain. The model suggests that a thalamocortical circuit might compute the direc- 
tional position of the rat's head, by integrating angular head velocity over time. 
4.1 COMPARISON WITH OTHER MODELS 
McNaughton et al. (1991) proposed that neurons encoding head-direction and angular 
velocity might be connected to form a linear associative mapping network. Skaggs et al. 
(1995) have refined this idea into a theoretical circuit, which incorporates head-direction 
and angular velocity cells. However, the Skaggs et al. (1995) circuit does not incorporate 
anticipatory head-direction cells, like those found in AT. A model that does incorporate 
anticipatory cells has been developed by Elga, Redish, & Touretzky (unpublished manu- 
script). Zhang (1995) has recently presented a theoretical analysis of the head-direction 
circuit, which suggests that anticipatory head-direction cells might be influenced by both 
the angular velocity and angular acceleration of the head, whereas non-anticipatory cells 
may be influenced by the angular velocity only, and not the angular acceleration. 
4.2 LIMITATIONS OF THE MODEL 
In its current form, the model suffers some significant limitations. For example, the direc- 
tional tuning curves of the modei's head-direction cells are much narrower than those of 
actual head-direction cells. Also, in its present form, the model can accurately track the 
rat's head-direction over a rather limited range of angular head velocities. These limita- 
tions are presently being addressed in a more advanced version of the model. 
158 H.T. BLAIR 
Acknowledgments 
This work was supported by NRSA fellowship number I F31 MHIlI02-01A1 from 
NIMH, a Yale Fellowship, and the Yale Neuroengineering and Neuroscience Center 
(NNC). I thank Michael Hines, Patricia Sharp, and Steve Fisher for their assistance. 
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