<title>1. INTRODUCTION</title> <p>The entorhinal cortex plays an important
role in memory function. In the rat,
entorhinal cortex lesions impair performance in both spatial memory tasks [##REF##15664181##1##] as well as in odor memory tasks [##REF##1445656##2##]. In monkeys, memory function
in delayed match to sample tasks is impaired by lesions of entorhinal cortex [##REF##7643207##3##] and adjacent parahippocampal structures [##REF##7874239##4##]. This article describes
cellular and circuit mechanisms in the entorhinal cortex that could underlie
its role in spatial and episodic memory functions. Computational modeling links data across multiple levels of function, including: (a) properties of single cell physiology, including persistent spiking and membrane potential oscillations, (b) properties of unit recording, including grid cells, place cells, and head
direction cells, and (c) the role of entorhinal cortex in spatial path
integration and episodic memory function. 
This article will review physiological data and modeling across these
different levels.</p>
<title>8. CONCLUSIONS</title> <p>The cellular physiological phenomena
described in this paper provide mechanisms important for behavioral functions
including path integration and the episodic encoding and retrieval of
trajectories. Detailed computational
models demonstrate the potential behavioral role of cellular mechanisms of
persistent spiking and membrane potential oscillation, demonstrate how these
could underlie unit recording data such as grid cell firing, and generate
predictions for future experimental studies.</p>
<p>Recommended by Roland S.G. Jones</p> <p>The entorhinal cortex plays an important role in spatial memory and episodic memory functions. These functions may result from cellular mechanisms for integration of the afferent input to entorhinal cortex. This article reviews physiological data on persistent spiking and membrane potential oscillations in entorhinal cortex then presents models showing how both these cellular mechanisms could contribute to properties observed during unit recording, including grid cell firing, and how they could underlie behavioural functions including path integration. The interaction of oscillations and persistent firing could contribute to encoding and retrieval of trajectories through space and time as a mechanism relevant to episodic memory.</p>
<title>2. PHYSIOLOGICAL DATA</title> <p>Recordings
from neurons in slice preparations of entorhinal cortex demonstrate important
cellular properties including (i) persistent spiking and (ii) membrane potential
oscillations. These cellular properties
could contribute to properties described in unit recordings from entorhinal
cortex in awake, behaving rats.</p> <title>2.1. Persistent spiking</title> <p>In slices, pyramidal neurons in different
layers of entorhinal cortex demonstrate the capacity to display persistent
spiking activity after a depolarizing current injection or a period of
repetitive synaptic input [##REF##9114238##5##–##REF##12432392##8##]. Pyramidal neurons in layer II of medial
entorhinal cortex show persistent spiking that tends to turn on and off over
periods of many seconds [##REF##9114238##5##]. This cyclical persistent spiking is shown in ##FIG##0##Figure 1(a)##. As described below, this could underlie the
spatial periodicity of grid cells. 
Pyramidal neurons in deep layers of entorhinal cortex can maintain
spiking at different graded frequencies for many minutes [##REF##12432392##8##] as shown in ##FIG##1##Figure 2(a)##. The
persistent spiking appears to due to muscarinic or metabotropic glutamate activation
of a calcium-sensitive nonspecific cation current [##REF##16504948##7##, ##REF##12205141##9##, ##UREF##0##10##]. This graded persistent
firing could allow these neurons to integrate synaptic input over extended
periods. Persistent firing has also been
shown in layer III of lateral entorhinal cortex [##REF##17315198##6##].</p> <title>2.2. Membrane potential oscillations</title> <p>Entorhinal layer II stellate cells show prominent subthreshold membrane
potential oscillations when depolarized near firing threshold [##REF##2812013##11##, ##REF##8395571##12##]. These are small oscillations of a few
millivolts in amplitude that can influence the timing of action potentials [##REF##15132436##13##] and can contribute to network oscillations [##REF##12843696##14##, ##REF##3653311##15##]. The frequency of membrane potential
oscillations differs systematically along the dorsal to ventral axis of the
medial entorhinal cortex [##REF##17379810##16##]. A model presented below discusses how the membrane potential
oscillation properties could underlie differences in grid cell firing
properties along the dorsal to ventral axis [##REF##17379810##16##–##UREF##1##19##]. The oscillations appear to
be due to a hyperpolarization activated cation current or h-current [##REF##10805658##20##], that differs in time constant along the dorsal to ventral axis [##UREF##2##21##]. Depolarizing input
increases the frequency of these oscillations such that the phase of the
oscillation integrates the depolarizing input over time. Membrane potential oscillations do not
usually appear in layer II or layer III pyramidal cells [##REF##8395571##12##], but are observed in layer V pyramidal cells, where they may be caused
by M-current [##REF##17728392##22##]. Membrane potential
oscillations do not appear in neurons of the lateral entorhinal cortex [##REF##16127693##23##].</p> <title>2.3. Unit recording data</title> <p>Recordings of neural activity in awake behaving
rats provide important clues to the functional role of entorhinal cortex. In particular, many cells in medial
entorhinal cortex fire as “grid cells.” 
A single grid cell responds as a rat forages in a hexagonal array of
different locations in an open-field environment [##REF##15965463##24##, ##REF##16675704##25##]. Examples of the pattern of firing in modeled
grid cells are shown in Figures ##FIG##0##1## and ##FIG##1##2##. 
Grid cells differ in spatial periodicity along the dorsal to ventral
axis of medial entorhinal cortex, with larger spacing between larger fields in
more ventral regions [##REF##15965463##24##, ##REF##16675704##25##].</p> <p>Grid
cells appear in all layers of entorhinal cortex, but in layers V and VI of
entorhinal cortex the grid cells often only respond when the rat is facing in a
particular direction [##REF##16675704##25##]. This resembles head
direction cells in areas such as the postsubiculum (dorsal presubiculum), which
respond at all locations in the environment but only when the rat faces a
particular direction [##REF##2303851##26##–##REF##8725961##30##]. The conjunctive grid-by-direction cells
resemble the theta-modulated place-by-direction cells observed in the post- and
parasubiculum, which respond only when the rat faces a preferred direction
while occupying a single location [##REF##15385610##31##].</p> <title>3. PATH INTEGRATION</title> <p>The cellular mechanisms described above may
contribute to the function of path integration. 
Path integration involves an animal using its self-motion cues to
maintain an accurate representation of the angle and distance from its start
position, even during performance of a complex trajectory through the
environment [##REF##16593958##32##–##REF##9317483##35##]. Many species demonstrate
the behavioral capacity to remember the distance as well as the angle of return
to the starting location (here represented in Cartesian coordinates by a two
component return vector). As shown here,
persistent firing provides a single neuron mechanism to integrate the distance
and angle of trajectory segments to compute the overall distance and angle from
start location to an end or goal location. 
Here, the vector from start to goal is called the goal vector (and the
negative of the goal vector is called the return vector). An example of a return vector is shown as a
dashed line in ##FIG##2##Figure 3(a)##.</p> <p>In general, the location of an agent can be
determined by computing the integral of the velocity vector. In Cartesian coordinates, the velocity vector
is . Integration of the velocity vector over a
period of time <italic>T</italic> after starting at
location  yields
the location vector  at time <italic>T</italic>:

</p> <p>For example, if the velocity of an animal is
10 cm/sec in the <italic>x</italic> direction and 5 cm/sec in the <italic>y</italic> direction, integration over
5 seconds yields a final location of [<italic>x</italic>, <italic>y</italic>] = [50 cm, 25 cm] relative to the start
location. Note that this integral
corresponds to the goal vector ,
which is the negative of the return vector.</p> <title>4. INTEGRATION BY PERSISTENT FIRING CAN CODE LOCATION</title> <p>Integration of the goal vector or return
vector could be provided by the mechanism of graded persistent firing in deep
layer entorhinal neurons [##REF##16504948##7##, ##REF##12432392##8##]. These neurons could
integrate a velocity vector coded by neurons responsive to the head direction
of the rat and to the speed of the rat. 
Head direction cells have been shown in deep entorhinal cortex [##REF##16675704##25##] and in the postsubiculum, which provides direct input to the
entorhinal cortex [##REF##2303851##26##, ##REF##8670654##27##]. Head direction cells respond selectively when
the rat is heading in a specific allocentric direction. Some neurons show sensitivity to speed
(translational motion) in the postsubiculum [##REF##8670654##27##] as well as in the hippocampus [##UREF##3##36##] and in the medial mammillary nucleus which receives output from the
postsubiculum and medial entorhinal cortex [##REF##15857969##37##]. Here, the activity of a
population of head direction cells modulated by speed is represented by
multiplying the velocity vector of the rat with a head direction matrix <italic>H</italic>. The head direction matrix consists of rows
with unit vectors representing the preference angles of individual speed
modulated head direction cells, that transforms the velocity vector into a head
direction vector . For example, a matrix representing two head
direction cells with preference angles <italic>θ</italic>
<sub>1</sub> and <italic>θ</italic>
<sub>2</sub> has two rows: . For a cell with a preference angle of 0, the
activity of the head direction cell would be cos(0)∗<italic>d</italic>
<italic>x</italic>/<italic>d</italic>
<italic>t</italic> + sin(0)<italic>d</italic>
<italic>y</italic>/<italic>d</italic>
<italic>t</italic> = <italic>d</italic>
<italic>x</italic>/<italic>d</italic>
<italic>t</italic>. This framework has the advantage
that it allows for a simple inverse transform turning head direction space back
into Cartesian coordinates. The inverse [##UREF##4##38##] of the matrix <italic>H</italic> is 

</p> <p>Note that this framework assumes that the
response of head direction cells is like a cosine function, whereas head
direction cells usually only show positive activity and have narrower,
triangular response functions with no activity outside this range. Head direction input corresponding to a
cosine function of actual input could be provided by summed input converging
from a population of head direction cells with different magnitudes of tuning
for values at different angles from the preferred direction. Note also that this representation combines
the properties of different neurons responding to translational velocity or to
head direction in the postsubiculum [##REF##8670654##27##] and other regions [##UREF##3##36##, ##REF##15857969##37##]. If head direction is not computed based on
velocity, then it could be integrated from angular velocity as a distinct
element of the state vector or could be computed based on angle to a different
reference point.</p> <p>Using
this mathematical representation of head direction input, the firing rate of a
set of graded persistent firing cells could integrate the input from a set of
head direction cells to yield a firing rate as follows:

where
vector  represents the firing rate of a population of
graded persistent firing cells at time <italic>T</italic>. For example, imagine two cells <italic>a</italic>
<sub>1</sub> and <italic>a</italic>
<sub>2</sub> with capacity for graded persistent spiking that receive
input from two head direction cells with preference angles of 0 degrees and 60
degrees. Imagine that the rat moves at
10 cm/sec in the <italic>x</italic> direction for 4 seconds, and the scaling factor <italic>β</italic> is 0.25 Hz/cm. Moving in the <italic>x</italic> direction is
equivalent to moving at 0 degrees, which would result in the head direction
cells having the activity <italic>h</italic>
<sub>1</sub> = 1 and <italic>h</italic>
<sub>2</sub> = 0.5. The
computation in (##FORMU##8##3##) would then result in the frequency of the graded persistent
firing cells as follows: . Thus, the graded persistent firing cells
would increase their activity to <italic>a</italic>
<sub>1</sub> = 10 Hz and <italic>a</italic>
<sub>2</sub> = 5 Hz. Mathematically, the inverse
transform of this firing rate vector computes the location vector  in Cartesian coordinates (see ##FIG##2##Figure 3(a)##).
However, the difference in neural activity could guide behavior without use of
the inverse transform. This could
involve forming associations between the start location and this pattern of
graded firing, and then forming associations between this pattern of graded
firing and the associated head direction signal. At the start location, the pattern of graded
firing could be activated, and this could retrieve the associated head direction signal. The animal could
change directions until its actual head direction matched this retrieved head
direction. This could give the animal
the correct angle to the goal.</p> <p>The
same mechanism computes both the goal vector and its negative, the return
vector. The return vector allows a rat
to return to the starting location from any arbitrary location in the
environment. In contrast, the goal
vector can be used to store the distance and angle to important locations in
the environment. For example, if a rat
is started in one location in an open field, and wanders until it finds food in
another location, the integrated activity vector at that point is the goal
vector—it provides a simple description of the angle
and distance from the start to the goal. 
This goal vector could be associated with all elements of the preceding
path by backward replay of place cells coding the full pathway [##REF##16474382##39##, ##REF##18242057##40##]. This could allow storage of an association
between place cells active at the start location and the subsequent goal
vector. Retrieval of the goal vector at
the start location could then allow the rat to go directly to the location of
food reward. If the spatial locations
leading to a goal are associated with the goal vector at each goal location,
and then integration is reset, a sequential series of trajectories to goals
could be stored separately. The rat
could then use this activity to sequentially retrieve pathways to different
rewarded locations in the environment, as in some behavioral tasks [##REF##7472463##41##]. The resetting of
integration activity could underlie the different pattern of place cell firing
shown with this type of directed task compared to open field activity. Thus, the resetting of integration could
explain the shift in firing location for place cells between scavenging in an
open field and following sequential trajectories between reward locations [##REF##7472463##41##] as well as the shift in firing location for grid cells between open
field scavenging and running on a long hairpin track [##UREF##5##42##] in which the view of each new segment could cause phase reset.</p> <p>Graded
persistent spiking could also hold initial head direction <italic>θ</italic>
<sub>HD</sub>(0) and update this by integrating input from
neurons coding angular head velocity  in areas such as the postsubiculum [##REF##8670654##27##].</p> <title>5. PERSISTENT SPIKING COULD UNDERLIE GRID CELL FIRING</title> <p>Because the neurons that show persistent
firing can integrate the synaptic input from speed modulated head direction
cells, and thereby can code spatial location, these persistent firing neurons
could potentially be the grid cells recorded in awake behaving animals [##REF##15965463##24##, ##REF##17322902##43##, ##REF##17486102##44##]. This section describes two potential
mechanisms for persistent firing neurons to contribute to the activity of grid
cells. The first mechanism involves the
cyclical persistent firing shown in layer II (see ##FIG##0##Figure 1(a)##). The second mechanism involves the graded
persistent firing shown in layer V (see ##FIG##1##Figure 2(a)##).</p> <p>In a general manner, the experimental data
on firing of single grid cells can be described by

where <italic>g</italic>(<italic>t</italic>) is the probability of
firing of the grid cell over time. The
product sign Π represents
multiplication of the output from each row (each head direction <italic><italic>θ</italic></italic>) of the head
direction transform matrix <italic>H</italic> described above. The description of the experimental data here
directly uses the vector representation of location over time <italic>x</italic>(<italic>t</italic>). Orientation of the grid is determined by the
head directions <italic><italic>θ</italic></italic> in <italic>H</italic>, the spatial phase is determined
by <italic>φ</italic>, and the spacing between fields is determined by the angular
frequency <italic>ω</italic>. This equation resembles other representations
of grid cells [##REF##17598147##18##, ##REF##17094145##45##] but simplifies the
representation by using the head direction transform matrix.</p> <title>5.1. Cyclical persistent firing</title> <p>The pattern of
periodic spatial firing of grid cells could arise from the pattern of cyclical
persistent firing as shown in ##FIG##0##Figure 1(a)##. 
The tendency for persistent firing to turn on and off could contribute
to grid cell firing if the oscillation could be gated by integration of input
from different populations of head direction cells with different preferred
angles. Simulations shown in Figures ##FIG##0##1(b)## and ##FIG##0##1(c)## show that the following equations can generate grid cell firing properties:
where <italic>h</italic>
<sup>+</sup> represents changes in current due to positive
components of the head direction input (note that this uses cosine modulated
head direction input), and <italic>h</italic>
<sup>−</sup> represents current due to negative
components of the cosine modulated head direction input. Note that the equations separately use
positive and negative elements of the speed modulated head direction matrix <italic>H</italic> transforming the rat movement velocity <italic>v</italic>(<italic>t</italic>). The parameter <italic>ω</italic> determines the time scaling of input effects
on activity levels. In the equations, <italic>V</italic>(<italic>t</italic>) represents the voltage change in individual compartments each of which receive
input from the positive and negative components of one head direction
input. As noted above, the cosine
modulated head direction input could be provided by summing over head direction
cells with different angles of preference. 
The negative influence of head direction in the equation could be due to
feedforward inhibition or inhibitory GABAergic projections from the
postsubiculum to the medial entorhinal cortex. 
The function <italic>g</italic>(<italic>t</italic>) represents the firing of grid cells over
time. The square brackets [] indicate
that spiking occurs whenever <italic>V</italic>(<italic>t</italic>) crosses a threshold.</p> <p>This pattern of activity could be obtained if neurons respond to
different head direction inputs with cyclical persistent firing (Klink and Alonso, 1997), as shown in ##FIG##0##Figure 1(a)##. 
In this case, when going one direction, head direction input will cause
phasic changes in firing in that direction, possibly due to build up first of
calcium and then of calcium inactivation. 
When going the exact opposite direction, head direction input would have
to activate the reverse processes, possibly reducing calcium inactivation and
then reducing calcium.</p> <p>Examples of grid field plots obtained with this model are shown in
Figures ##FIG##0##1(b)## and ##FIG##0##1(c)##. The grid fields
are more stable in the trajectory data from the Moser laboratory than in
randomly created trajectories (see ##FIG##0##Figure 1(b)##) or in a trajectory obtained in
our own laboratory (see ##FIG##0##Figure 1(c)##). 
This indicates that the statistics of rat movement can determine
appearance of the grid in this new model, and this could underlie variability
in detection of grid cell firing properties depending on the trajectory of rat
movement in the behavioral foraging task.</p> <title>5.2. Graded persistent firing</title> <p>As an alternative to cyclical persistent firing, graded persistent firing as shown in ##FIG##1##Figure 2(a)## could provide the basis for grid cell firing. In this framework, different graded persistent firing cells start out with the same baseline frequency of spiking and provide convergent input to a grid cell that fires whenever the inputs are
in synchrony. Speed modulated head
direction input to different graded persistent firing cells will transiently alter the
frequency and thereby the phase of firing. 
Therefore, if the rat moves, it shifts the frequency of a graded cell
out of phase with the other cells and thereby reduces or stops the grid cell
firing until the phase is shifted enough to come into phase with the other
neurons. A grid cell simulated with this
model is shown in ##FIG##1##Figure 2(b)##. This
mechanism uses graded persistent firing in a manner similar to the mechanism of
membrane potential oscillations described in the following section.</p> <p>Both of these models will yield a pattern of firing similar to grid
cells as long as the head direction cells providing input have preference
angles at multiples of 60 degrees. For
path integration, the head direction angles used for integration might be
determined at the start location. For
example, a single pyramidal cell showing persistent firing might receive input
from three head direction cells that code the heading angle at the start, as
well as the angle of eye direction. Rats
have binocular overlap of about 60 degrees [##UREF##6##46##]. If the total visual field of one eye is 180 degrees and the center
of view is at 90 degrees, then the center of view for each eye should be offset
about 60 degrees from head direction. 
Therefore, a rat may choose angles of 0, −60, and 60 degrees for path
integration. These angles have the 60-degree
difference necessary for the head direction input to cause hexagonal arrays in
the grid cell model. The rat can use
these initial angles of view as reference angles, but if it turns far enough
from the initial heading (e.g., 180 degrees from the initial head direction),
then it may need to select additional reference angles at 60-degree intervals
from the previous reference angles.</p> <p>Some grid cells respond selectively only for certain head direction [##REF##16675704##25##]. These head direction
sensitive grid cells might result from the input only being suprathreshold for
a population of head direction cells responding near one preferred angle, with
input being subthreshold from other populations of head direction cells coding
other preferred angles (e.g., at 60 or 120 degrees differences).</p> <title>6. INTEGRATION BY MEMBRANE POTENTIAL OSCILLATION PHASE CAN CODE LOCATION</title> <p>As an alternative mechanism for path
integration, the phase of membrane potential oscillations in medial entorhinal
stellate cells can also be used to integrate speed modulated head direction
input. This mechanism was proposed in a
model of grid cells developed by Burgess et al. [##REF##17598147##18##] and O'Keefe and Burgess [##REF##16145693##47##]. This mechanism uses the physiological fact
that depolarizing inputs to stellate cells cause a change in frequency of
membrane potential oscillations [##REF##17379810##16##]. This could change
oscillation phase based on an integral of the depolarizing input.</p> <title>6.1. Model of grid cells using membrane potential oscillations</title> <p>This
computational model shows how activity of a single grid cell could arise from
membrane potential oscillations within that cell modulated by depolarizing
input from head direction cells:

where <italic>g</italic>(<italic>t</italic>) is the firing in time and space of a single modeled grid cell. <italic>ω</italic> represents the baseline angular frequency of membrane
potential oscillations (2∗<italic>π</italic>∗<italic>f</italic>) in different
portions of the neuron. <italic><italic>β</italic><sub>H</sub></italic> is the experimentally determined scaling factor relating membrane
potential oscillations to grid cell spacing. 
The input from head direction cells is determined by the matrix <italic>H</italic> and the velocity vector . The inner product of each row of <italic>H</italic> with the
velocity vector  represents input to one dendrite from one head
direction cell modulated by the speed of the rat. This input alters the frequency of dendritic
membrane potential oscillations and thereby shifts the phase of the dendritic
oscillations in proportion to the integral of the velocity vector over
time. Both the starting location of the
rat and the spatial phase of the grid cell are combined in the initial location
vector <italic>x</italic>
<sub>0</sub>. This initial location vector is also transformed
by the matrix <italic>H</italic>. The square brackets [] represent a Heaviside step
function generating a spiking output for each time point when the product
crosses a threshold (set at 1.8).</p> <p>This
model generates grid cell firing fields with spacing between fields dependent
upon the frequency of membrane potential oscillations [##REF##17379810##16##–##REF##17598147##18##]. A grid cell created with
this model is shown in ##FIG##1##Figure 2(c)##. The
model generated the prediction that the systematic change in spacing of grid
cell firing fields along the dorsal to ventral axis of entorhinal cortex would
depend upon a systematic difference in frequency of membrane potential
oscillations in entorhinal neurons. This
prediction was tested and supported by whole cell patch recordings from
entorhinal layer II stellate cells [##REF##17379810##16##]. Based on experimental data
alone, it appears that membrane potential oscillation frequency <italic>f</italic> is
scaled to grid cell spacing <italic>G</italic> by a constant factor <italic>f</italic> ∗ <italic>G</italic> = <italic>H</italic> [##REF##17379810##16##, ##REF##17924530##17##].</p> <p>As noted above, membrane potential
oscillations appear in specific medial entorhinal populations such as layer II
stellate cells and layer V pyramidal cells, but not in other cells such as
layer II pyramidal cells, or neurons in medial entorhinal layer III, or in the
lateral entorhinal cortex. Based on
these data, the generation of grid cells responses based on membrane potential
oscillations would only occur in layer II stellate cells and layer V pyramidal
cells, and would appear in other neurons due to network interactions or due to
the persistent spiking mechanisms described above.</p> <title>6.2. Grid cell activity codes location</title> <p>In the model, the depolarizing input from head direction cells increases the frequency of membrane potential oscillations in proportion to the velocity
vector. This shift in frequency alters
the phase of oscillations in proportion to the integral of the velocity vector
transformed by head direction: . In addition, the interference pattern
increases and decreases in proportion to the difference in oscillation
frequency of the soma and dendrite, so that the vector of angular phases of
interference is  (see Figures ##FIG##2##3(b)## and ##FIG##2##3(c)##) and the equation
for grid cell activity can be written for location as 
</p> <p>The location can be extracted from grid cell
phase by using the inverse of the head direction transform matrix as follows: . ##FIG##2##Figure 3(d)## shows the trajectory obtained from
this inverse transform of phase.</p> <title>6.3. Theta phase precession</title> <p>In
addition to replicating the spatial periodicity of grid cell firing fields, the
model based on interference of membrane potential oscillations also replicates
experimental data showing systematic changes in phase of grid cell firing
relative to network theta rhythm oscillations [##UREF##7##48##], a phenomenon known as theta phase precession. The phenomenon of theta phase precession was
initially shown for place cell firing in the hippocampal formation [##REF##8353611##49##, ##REF##8797016##50##] and was proposed to
arise from the interaction of network theta rhythm oscillations with cellular
theta rhythm oscillations [##REF##8353611##49##, ##REF##16145692##51##, ##REF##12962315##52##]. As an alternative model, precession was
proposed to arise from the readout of sequences of place cell activity [##REF##8841826##53##–##REF##10456097##55##].</p> <p>The oscillatory interference model of grid
cells [##REF##17598147##18##, ##REF##16145693##47##] extended the models of
hippocampal phase precession and can account for grid cell phase precession [##UREF##7##48##]. In the oscillatory
interference model, the interference has two components that appear
physiologically: (1) as described above, the size of grid fields is determined
by the envelope that has a frequency depending on the difference of the two
angular frequencies <italic>ω</italic>
<sub><italic>d</italic></sub> − <italic>ω</italic>
<sub><italic>s</italic></sub>, and (2) the pattern of phase precession depends on a higher
frequency component that corresponds to the mean of the two frequencies (<italic>ω</italic>
<sub><italic>s</italic></sub> + <italic>ω</italic>
<sub><italic>d</italic></sub>)/2. The phase of this
second-high frequency component of the summed oscillation is . The spiking will occur near the peak phase of
this summed oscillation which is equal to some multiple <italic>n</italic> of the full
cycle: <italic>φ</italic>
<sub>sum</sub> = <italic>n</italic>2<italic>π</italic>.</p> <p>The spiking caused by the phase of the
summed oscillations can then be plotted relative to the network theta rhythm by
plotting the phase of the soma at the time of each spike. This can be obtained analytically from the
above equation if we consider the case of the movement at a constant speed
continuously in the preferred direction of one head direction cell. In this case, the integral of head direction
for that cell is simply the integral of speed, which is equal to the location . Note also that the phase of the soma
oscillations is the product of soma frequency and time: <italic>φ</italic>
<sub>soma</sub> = 2<italic>π</italic>
<italic>f</italic>
<italic>t</italic>. Therefore, the equation for the phase of the
summed oscillation can be reduced to <italic>φ</italic>
<sub>sum</sub> = <italic>φ</italic>
<sub>soma</sub> + <italic>ω</italic>
<italic>B</italic>
<italic>x</italic>/2.</p> <p>Plotting of theta phase precession
essentially involves plotting the timing of spikes (which occur when <italic>φ</italic>
<sub>sum</sub> = <italic>n</italic>2<italic>π</italic>) relative to the phase of the network
oscillations (which here correspond to the phase of the soma because the soma
is being driven by network oscillations with fixed frequency <italic>ω</italic>). Thus, the vertical axis of a plot of theta
phase precession shows the phase of the soma oscillation at the time of each
spike: <italic>φ</italic>
<sub>soma</sub> = <italic>n</italic>2<italic>π</italic> − <italic>π</italic>
<italic>f</italic>
<italic>B</italic>
<italic>x</italic> plotted relative to location <italic>x</italic> on the
horizontal axis. ##FIG##3##Figure 4## shows the
plotting of spikes in the simulation during runs on a linear track through the
firing field of the neuron. Note that
the phase precession in this model resembles the phase precession found in experimental data [##REF##8797016##50##] but only covers about 180 degrees of the network theta oscillation
cycle.</p> <p>Note that the phase precession for a single
direction gives a partial readout of the phase code of location, but when
considering the phase in two-dimensional space, it confounds the phase of the two or more
dendrites, so it is radially symmetric and dependent upon the direction of
trajectory through the field (see ##FIG##4##Figure 5(a)##). Thus, the phase
precession code is less accurate for use in path integration, in contrast to
the overall mean firing rate that would be observed in a grid cell due to
persistent firing with a very large firing field, which could code location for
distances smaller than one half the spacing between two grid fields (e.g., for 80 cm spacing,
distances less than 40 cm could be coded).</p> <title>6.4. Mechanism for path integration</title> <p>As
noted above, path integration involves an animal using its self-motion cues to
maintain an accurate representation of angle and distance from start
position. In the grid cell model, the
update of oscillation phase by speed-modulated head direction integrates the
velocity vector, thereby integrating the distance from the starting position in
specific directions determined by <italic>H</italic>. 
If oscillation phases are reset to zero at the starting location, the
return vector <bold>r</bold> giving distance and direction back to the starting location
can be obtained by applying the inverse of the head direction matrix to the
dendritic phase vector at any position . ##FIG##2##Figure 3(a)## shows that after resetting phase at the
starting location, applying the inverse head direction transform to the dendritic phase vector, and taking the negative of this vector gives the direction and angle directly back
to the starting location (dotted line).</p> <title>6.5. Interaction of path integration and visual stimuli</title> <p>Path
integration based on idiothetic cues alone builds up substantial error [##REF##17520273##34##] that is usually corrected by comparison with sensory cues such as
visual stimuli [##REF##9317483##35##]. Grid cells show dependence
upon visual stimuli in the environment. 
They maintain the same properties when returned to a familiar room [##REF##15965463##24##], and they change their scale during a period of time after
manipulations of environment size [##REF##17486102##44##]. This influence of visual
stimuli could result from the fact that the angle to visual stimuli has the
same properties as phase of grid cell oscillations. ##FIG##4##Figure 5## shows how the angle of a single
distal visual stimulus changes as a rat moves in a manner that is consistent
with the change in phase of individual dendrites of a single modeled grid cell.</p> <p>Alternately, the grid cell could be
influenced by the angle and distance to visual stimuli, and neural mechanisms
could update the expected angle and distance to visual stimuli in a manner
similar to the mechanism for updating the angle and distance from start
location (return vector). This requires
updating the angles of the initially selected stimuli (that might be determined
by eye direction) according to the direction and velocity of movement. This will update the expected absolute angle
of visual stimuli. The further
computation of expected relative angle (the actual visual input) requires
combining absolute angle with current head direction. The process of updating head direction could
depend upon input from cells coding angular velocity of movement. These have been shown in the postsubiculum [##REF##8670654##27##, ##UREF##8##56##] as well as in structures
including the anterior dorsal thalamic nucleus [##REF##9643555##29##]. Grid cells appear to be
more consistent when there are clear barriers on the edge of the open field,
suggesting that rats might use the vertical angle of a boundary to judge
distance.</p> <p>The basic grid cell model assumes speed
modulation of head direction cells, but most head direction cells show stable
persistent firing even when the rat is motionless. In contrast, place cells show more speed
modulation. In keeping with the
physiological data that shows stable persistent firing for head direction cells
and speed dependent firing for place cells, it might be appropriate to
represent state as the static head direction cell activity combined with visual
stimulus angle, and to use the speed-dependent activity of place cells as the
action of the rat.</p> <title>6.6. Grid cell phase represents continuum of locations for reinforcement
learning</title> <p>The grid cells can be used as a representation of state for goal
directed behavior. Many reinforcement
learning theory models have used discrete representations of state for goal
directed behavior. However, this causes
difficulties for representing movement in continuous space. The phase of oscillations in the grid cell
models is a continuous representation of space that can be updated in a continuous
manner by actions held by persistent spiking. 
As noted previously [##REF##17379810##16##, ##REF##18007021##57##], this could allow grid
cells to provide an effective mechanism for representations of state and action
in continuous space and time.</p> <title>7. MODEL OF EPISODIC MEMORY</title> <p>The
interaction of head direction cells and grid cells described here provides a
potential mechanism for episodic memory involving the storage of trajectories
through space and time [##REF##18007021##57##]. As shown in ##FIG##5##Figure 6##, this
model uses a functional loop that encodes and retrieves trajectories via three
stages: (1)
head direction cells <italic>h</italic>(<italic>t</italic>) update grid cells, (2) grid cells <italic>g</italic>(<italic>t</italic>) update place cells, and (3) place cells <italic>p</italic>(<italic>t</italic>) activate associated head direction
activity [##REF##18007021##57##]. This model is consistent
with the anatomical connectivity (see
##FIG##5##Figure 6##). The
head direction cells could update grid cells via projections from the
postsubiculum (dorsal presubiculum) to the medial entorhinal cortex [##REF##1704281##58##–##REF##3902916##60##], causing updating of persistent firing as described above, or
influencing the phase of membrane potential oscillations [##REF##17379810##16##–##REF##17598147##18##]. Grid cells can update place
cells via the extensive projections from entorhinal cortex layer II to dentate gyrus
and CA3 and from layer III to region CA1 [##REF##17094145##45##, ##REF##2687721##61##]. The connectivity from grid cells to place
cells could be formed by different computational mechanisms [##REF##17094145##45##, ##REF##16624947##62##, ##REF##17162463##63##]. Place cells can become associated with head
direction activity via direct projections from region CA1 to the postsubiculum [##REF##1704281##58##], or via indirect projections from region CA1 to the subiculum [##REF##2687721##61##, ##REF##690280##64##], and projections from
the dorsal and distal regions of the subiculum to the postsubiculum and medial
entorhinal cortex [##REF##9548550##65##], both of which contain head direction cells.</p> <p>During initial encoding of a trajectory in
the model, the head direction cell activity vector would be set by the actual
head direction of the rat during exploration, and associations would be encoded
between place cell activity and head direction activity. These associations would be stored in the
form of a synaptic connectivity matrix <italic>W</italic>
<sub>HP</sub> with strengthened
connections between active place cells <italic>p</italic>(<italic>t</italic>) and the head direction cell
activity vector <italic>h</italic>(<italic>t</italic>) as follows:</p> <p>In this equation, head direction vectors
associated with individual place cell locations are identified with the place
cell index <italic>p</italic>. During retrieval,
the head direction activity depends upon synaptic input from place cell
representations as follows: <italic>h</italic>(<italic>t</italic>) = <italic>W</italic>
<sub>HP</sub>
<italic>p</italic>(<italic>t</italic>).</p> <p>This model has the capacity for performing
episodic encoding and retrieval of trajectories in simulations [##REF##18007021##57##], including trajectories based on experimental data or trajectories
created by an algorithm replicating foraging movements of a rat in an open
field [##REF##17924530##17##]. During encoding, a series
of place cells <italic>p</italic>(<italic>t</italic>) is
created associated with particular locations <italic>x</italic>
<sub><italic>p</italic></sub> = <italic>x</italic>(<italic>t</italic>). Each place cell is also associated with input
from the grid cell population activity <italic>g</italic>(<italic>t</italic>) and with the head direction vector <italic>h</italic>
<sub><italic>p</italic></sub> = <italic>h</italic>(<italic>t</italic>) that occurred during the initial movement from
that location. For retrieval, the
simulation is cued with the grid cell phase vector <italic>φ</italic>(<italic>t</italic>
<sub>0</sub>) and head direction vector <italic>h</italic>(<italic>t</italic>
<sub>0</sub>) present at the start location. The head
direction vector updates the grid cell phase vector <italic>φ</italic>
<sub><italic>d</italic></sub>(<italic>t</italic>),
which alters the activity of grid cells. 
The grid cell firing drives place cells <italic>p</italic> associated with
subsequent locations on the trajectory. 
These mechanisms are summarized in ##FIG##5##Figure 6##.</p> <p>The activation of each new place cell
activates a new head direction vector <italic>h</italic>
<sub><italic>p</italic></sub> associated
with that place cell. This new head
direction vector then drives the further update of dendritic phases of grid
cells. This maintenance of the head direction vector might require graded
persistent spiking [##REF##12432392##8##] of head direction cells in deep layers of entorhinal cortex. Essentially, the retrieval of the place cell
activity representing the state drives the retrieval of the new head direction
vector representing the action from that state. 
This action is then used for a period of time to update the grid cell
state representation until a new place cell representation is activated.</p> <p>Because retrieval of the trajectory depends
on updating of phase by head direction cells, this allows retrieval of a
trajectory at a time course similar to the initial encoding. This can allow effective simulation of the
slow time course of place cell replay observed in previous experimental data
collected during REM sleep [##REF##11182087##66##]. The spread of activity from
place cells to cells coding head direction could contribute to patterns of
firing in the postsubiculum that appear as cells responding dependent on both
place and head direction [##REF##15385610##31##]. These cells might code the
action value for retrieval of a trajectory from a particular location, firing
only when actual head direction matches the head direction previously
associated with specific place cell activity. 
The strong theta phase specificity of these cells could be due to
separate dynamics for encoding and retrieval within each cycle of theta rhythm [##REF##11936962##67##]. These cells might
selectively fire during the retrieval phase.</p> <title>7.1. Enhancement by arc length cells</title> <p>The retrieval mechanism
mediated by place cells can be enhanced by inclusion of cells that fire
dependent upon the arc length of the trajectory [##REF##18007021##57##], or by the time interval alone [##REF##17924530##17##]. These types of responses
help prevent a breakdown in trajectory retrieval caused by overlaps in the
trajectory. The associations between
place cells and head direction cells cannot disambiguate between two segments
of a trajectory passing through the same location with different head directions. However, coding of arc length or time since
the start of the trajectory can disambiguate these overlapping locations. Oscillatory interference between neurons
directly modulated by speed but not head direction can activate arc length
cells coding arc length from the start of a trajectory, or from the last time
that oscillations are reset along the trajectory. Simulations based on this coding of arc
length can account for many features of unit recording data in behavioral
tasks. Persistent spiking in layer III of entorhinal cortex could provide a
means for driving the coding of arc length (or time) along a trajectory. Persistent spiking in layer III with specific
frequencies [##REF##17315198##6##] could activate neurons in region CA1 with different phases relative
to arc length (or time) on a trajectory. 
During retrieval, arc length cells from one portion of a trajectory can
activate associated speed modulated
cells to trigger the next arc length cell along a trajectory. This retrieval process can be accelerated or
decelerated via modulation of the frequency of entorhinal oscillations during
persistent firing.</p> <title>7.2. Predictions of arc length model</title> <p>Since the output of arc length cells is essentially dependent upon
distance, a simple manipulation of running distance should directly influence
spatial specificity of arc length cells. 
For example, in a rat running continuously clockwise around a circular
track with a circumference of two meters, an arc length cell may match the
periodicity of the track and display a stable place field somewhere on the track. Here, the cell is firing at an arc length of
two meters. If we expand the track by a
small amount to say, a circumference of 2.1 meters, the arc length cell would
be expected to continue to fire at two-meter intervals, thus the field will translocate
in the counterclockwise direction (or backwards in relation to the rat) by 10 centimeters
for each lap.</p> <p>The reset version of the arc length
coding model assumes that oscillations reset at a specified location or during
a key event such as food reward. This is
supported experimentally given that the theta oscillation does reduce when an
animal stops or consumes food. By using
the same manipulation on the circular track as before, a similar but quite
different prediction surfaces. Here, the
field will shift counterclockwise abruptly, but will remain at that location
for subsequent laps. This location
stability is a direct consequence of stability of the food reward location
since now the oscillatory interference is anchored to the food reward, and not
from the previous location that the cell had discharged. Interestingly, the
distance a field moves will be linearly proportional to the distance the
original field was from the food reward location. Thus, a field will move 10
centimeters only if the field was originally located at the end of a lap (at the
feeder), and a field will shift 5 centimeters if the original field was located
at the midpoint of the lap.</p> <p>The reset model prediction of the expanded
circular track leads us to a further prediction. Since the discharge of an arc length cell in
the reset model is dependent on and anchored to the reward location, a
manipulation of the location of food reward will cause a relative movement of
the location of the arc length's discharge. 
For example, the movement of the food reward by 10 centimeters in the
clockwise directly will cause an arc length cell to correspondingly shift its
field 10 centimeters in the clockwise direction.</p>
<title>ACKNOWLEDGMENTS</title> <p>Research supported by Silvio O. Conte Center grants NIMH MH71702 and MH60450, by NIMH R01 grants MH60013 and MH61492, by NSF SLC SBE 0354378, and by NIDA grant DA16454 (part of the CRCNS program).</p>
<fig id="fig1" position="float"><label>Figure 1</label><caption><p>(a) Example of persistent
firing in layer II pyramidal cell showing alternating cycles of spiking and nonspiking in data from Klink and Alonso [##REF##9114238##5##]. (b) Simulation of grid
cell firing dependent upon cyclical persistent spiking gated by random movement
in a two-meter square environment. Spiking shown as black dots on trajectory in gray. (c) Simulation from same model
using rat trajectory from experimental data in an 85 cm square
environment.</p></caption></fig> <fig id="fig2" position="float"><label>Figure 2</label><caption><p>(a) Example of graded
persistent firing in a layer V pyramidal cell from Egorov et al. [##REF##12432392##8##]. (b) Simulation of grid
cell firing based on persistent firing in cells from deep layers of
medial entorhinal cortex. The spiking
activity shown as black dots arises from convergent input from three neurons
with the same baseline persistent firing frequency, with phase of input neurons
influenced by input from different speed modulated head direction cells during
movement (trajectory shown in gray). (c)
Simulation of grid cell firing based on membrane potential oscillations in
dorsal layer II stellate cells in medial entorhinal cortex.</p></caption></fig> <fig id="fig3" position="float"><label>Figure 3</label><caption><p>Coding of location by phase. (a) Actual trajectory run by the rat is shown
in gray. If phase is reset at start
location, the inverse transform of phase at any position yields a return vector
with angle and distance leading back to start (shown with black dashed
line). (b) Plot of membrane potential
oscillation phase <italic>φ</italic> in a single dendrite
of a simulated grid cell, showing linear change in phase with one dimension of location (dendrite receives input from head direction cell with angle preference zero). (c) Phase of another dendrite receiving input from head direction with angle preference 120. (d) Performing the inverse transform of oscillation phases at each point in time effectively reconstructs the full
trajectory.</p></caption></fig> <fig id="fig4" position="float"><label>Figure 4</label><caption><p>Simulation of theta phase precession in grid cell model based on
membrane potential oscillations. (a)
Theta phase plotted on vertical axis as a simulated rat runs through a grid
cell firing field in west to east and east to west directions. (b) Theta phase during run from south to
north. (c) Spike times (filled circles)
of summed oscillations in a neuron relative to oscillation in the soma of that
neuron (negative of network theta oscillation).</p></caption></fig> <fig id="fig5" position="float"><label>Figure 5</label><caption><p>Phase relative to location in the
environment. (a) Spiking phase due to
precession (with refractory period). 
Note that phase depends upon location, but is circularly symmetric. (b) Dendritic phase of oscillations contains
more complete continuous representation of location. (c) Plot of the angle of a single distal
visual stimulus as a rat moves around in an environment, indicating similarity
of allocentric stimulus angle to integrated dendritic phase in a grid cell.</p></caption></fig> <fig id="fig6" position="float"><label>Figure 6</label><caption><p>Model of episodic encoding
and retrieval of trajectories. Top:
Schematic representation of connectivity between grid cells, place cells, and
head direction (HD) cells. Bottom: example
of trajectory retrieval activity in each region. Trajectory experienced during encoding is
shown in gray. (a) The location coded by
the oscillation phase of entorhinal grid cell membrane potential is plotted as
a dashed line that follows the actual trajectory in gray. Grid cell phase is put through the inverse
head direction transformation to obtain coded location. Phase shifts are driven by retrieved head
direction until next place cell is activated, then phase moves in new direction
dependent on next active head direction cell. 
(b) Sequentially activated place cell representations are shown as open
circles. (c) Each place cell activates a
corresponding head direction representation, with direction shown as a short,
straight, black line. This drives the
next period of update of the grid cell phase.</p></caption></fig>
<inline-formula><mml:math id="M1"><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>⇀</mml:mo></mml:mover><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>y</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M2"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>⇀</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M3"><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>⇀</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></inline-formula> <disp-formula id="eq1"><label>(1)</label><mml:math id="M4"><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>⇀</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>⇀</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:mstyle><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mi>T</mml:mi></mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>⇀</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:math>.</disp-formula> <inline-formula><mml:math id="M5"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M6"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M7"><mml:mrow><mml:mi>H</mml:mi><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mo minsize="1em" maxsize="1em">[</mml:mo><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mtext> </mml:mtext><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mtd><mml:mtd><mml:mi>sin</mml:mi><mml:mo>⁢</mml:mo><mml:mtext> </mml:mtext><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>cos</mml:mi><mml:mo>⁢</mml:mo><mml:mtext> </mml:mtext><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi>sin</mml:mi><mml:mo>⁢</mml:mo><mml:mtext> </mml:mtext><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo minsize="1em" maxsize="1em">]</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> <disp-formula id="eq001"><label>(2)</label><mml:math id="M8"><mml:mrow><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mtd><mml:mtd><mml:mo>−</mml:mo><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>−</mml:mo><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mrow><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable><mml:mo>]</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mo minsize="1em" maxsize="1em">(</mml:mo><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>cos</mml:mi><mml:mo>⁢</mml:mo><mml:mtext> </mml:mtext><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo minsize="1em" maxsize="1em">)</mml:mo><mml:mn>.</mml:mn></mml:mrow></mml:math></disp-formula> <disp-formula id="eq2"><label>(3)</label><mml:math id="M9"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mi>β</mml:mi><mml:mstyle><mml:mrow><mml:msubsup><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:mstyle><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mi>β</mml:mi><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>−</mml:mo><mml:mo> </mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> <inline-formula><mml:math id="M10"><mml:mrow><mml:mover accent="true"><mml:mi>a</mml:mi><mml:mo>⇀</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M11"><mml:mrow><mml:mi>a</mml:mi><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mi>β</mml:mi><mml:mrow><mml:mo minsize="1em" maxsize="1em">[</mml:mo><mml:mtable><mml:mtr><mml:mtd><mml:mi>cos</mml:mi><mml:mo>⁢</mml:mo><mml:mtext> </mml:mtext><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi>sin</mml:mi><mml:mo>⁢</mml:mo><mml:mtext> </mml:mtext><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>cos</mml:mi><mml:mo>⁢</mml:mo><mml:mtext> </mml:mtext><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi>sin</mml:mi><mml:mo>⁢</mml:mo><mml:mtext> </mml:mtext><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo minsize="1em" maxsize="1em">]</mml:mo></mml:mrow><mml:mrow><mml:mo minsize="1em" maxsize="1em">[</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mi>x</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>y</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo minsize="1em" maxsize="1em">]</mml:mo></mml:mrow><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mn>0.25</mml:mn><mml:mrow><mml:mo minsize="1em" maxsize="1em">[</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mn>1</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mn>0.5</mml:mn></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mn>0.87</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo minsize="1em" maxsize="1em">]</mml:mo></mml:mrow><mml:mrow><mml:mo minsize="1em" maxsize="1em">[</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>40</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo minsize="1em" maxsize="1em">]</mml:mo></mml:mrow><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mo minsize="1em" maxsize="1em">[</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>5</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo minsize="1em" maxsize="1em">]</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M12"><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>⇀</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mover accent="true"><mml:mi>a</mml:mi><mml:mo>⇀</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>β</mml:mi></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M13"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mtext>AHV</mml:mtext></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> <disp-formula id="eq3"><label>(4)</label><mml:math id="M14"><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mstyle><mml:mrow><mml:munder><mml:mrow><mml:mo>∏</mml:mo></mml:mrow><mml:mrow><mml:mi>θ</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>ω</mml:mi><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:mi>φ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> <disp-formula id="eq4"><label>(5)</label><mml:math id="M15"><mml:mrow><mml:mtable width="auto" columnalign="left"><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mo>ⅆ</mml:mo><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mtd><mml:mtd><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>V</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>H</mml:mi><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mo>ⅆ</mml:mo><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msup><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>V</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>H</mml:mi><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo minsize="0.75em" maxsize="0.75em">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mo>ⅆ</mml:mo><mml:msup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow/></mml:msup></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>H</mml:mi><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mrow/><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>H</mml:mi><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd><mml:mtd><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mo>∏</mml:mo><mml:mi>V</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">]</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula> <disp-formula id="eq5"><label>(6)</label><mml:math id="M16"><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mo minsize="1.5em" maxsize="1.5em">[</mml:mo><mml:mstyle><mml:mrow><mml:mo>∏</mml:mo><mml:mrow><mml:mo minsize="1.5em" maxsize="1.5em">(</mml:mo><mml:mi>cos</mml:mi><mml:mo>⁢</mml:mo><mml:mtext> </mml:mtext><mml:mo>⁡</mml:mo><mml:mi>ω</mml:mi><mml:mi>t</mml:mi><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mo minsize="1.5em" maxsize="1.5em">(</mml:mo><mml:mi>ω</mml:mi><mml:mi>t</mml:mi><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:mi>ω</mml:mi><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:msub><mml:mstyle><mml:mrow><mml:msubsup><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mtext> </mml:mtext><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mi>d</mml:mi><mml:mi>τ</mml:mi><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo minsize="1.5em" maxsize="1.5em">)</mml:mo><mml:mo minsize="1.5em" maxsize="1.5em">)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo minsize="1.5em" maxsize="1.5em">]</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> <inline-formula><mml:math id="M17"><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>⇀</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M18"><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>⇀</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M19"><mml:mrow><mml:mstyle><mml:mrow><mml:msubsup><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mstyle><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M20"><mml:mrow><mml:mi>φ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mi>ω</mml:mi><mml:mi>β</mml:mi><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M21"><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mstyle displaystyle="false"><mml:mo>∏</mml:mo><mml:mrow><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>ω</mml:mi><mml:msub><mml:mi>β</mml:mi><mml:mi>H</mml:mi></mml:msub><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>⇀</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle><mml:mo stretchy="false">]</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M22"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mover accent="true"><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>ω</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M23"><mml:mrow><mml:msub><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mtext>sum</mml:mtext></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mi>ω</mml:mi><mml:mi>t</mml:mi><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:mi>ω</mml:mi><mml:mi>β</mml:mi><mml:mstyle><mml:mrow><mml:msubsup><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mtext> </mml:mtext><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>d</mml:mi><mml:mi>τ</mml:mi></mml:mrow></mml:mrow></mml:mstyle><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M24"><mml:mrow><mml:mi>x</mml:mi><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mstyle><mml:mrow><mml:msubsup><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mi>H</mml:mi><mml:mover accent="true"><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mi>d</mml:mi><mml:mi>τ</mml:mi></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M25"><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mover accent="true"><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo>/</mml:mo><mml:mi>ω</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:math></inline-formula> <disp-formula id="eq6"><label>(7)</label><mml:math id="M26"><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mtext>HP</mml:mtext></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msup><mml:mrow><mml:mstyle><mml:mrow><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>⇀</mml:mo></mml:mrow></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:mstyle></mml:mrow><mml:mrow/></mml:msup><mml:mn>.</mml:mn></mml:mrow></mml:math></disp-formula>
<graphic xlink:href="NP2008-658323.001"/> <graphic xlink:href="NP2008-658323.002"/> <graphic xlink:href="NP2008-658323.003"/> <graphic xlink:href="NP2008-658323.004"/> <graphic xlink:href="NP2008-658323.005"/> <graphic xlink:href="NP2008-658323.006"/>
[{"label": ["10"], "surname": ["Yoshida", "Frans\u00e9n", "Hasselmo"], "given-names": ["M", "E", "ME"], "article-title": ["Cholinergic-independent persistent firing in entorhinal layers III and V neurons"], "italic": ["Society for Neuroscience Abstract"], "year": ["2007"], "volume": ["33"], "fpage": ["935.9 pages"]}, {"label": ["19"], "surname": ["Burgess", "Barry", "Jeffery", "O'Keefe"], "given-names": ["N", "C", "KJ", "J"], "article-title": ["A grid and place cell model of path integration utilizing phase precession versus theta"], "conf-name": ["In: Proceedings of the 1st Annual Computational Cognitive Neuroscience Conference"], "conf-date": ["April 2005"], "conf-loc": ["Washington, DC, USA"]}, {"label": ["21"], "surname": ["Giocomo", "Hasselmo"], "given-names": ["LM", "ME"], "article-title": ["Time constant of I(h) differs along dorsal to ventral axis of medial entorhinal cortex"], "comment": ["submitted"]}, {"label": ["36"], "surname": ["O'Keefe", "Burgess", "Donnett", "Jeffery", "Maguire"], "given-names": ["J", "N", "JG", "KJ", "EA"], "article-title": ["Place cells, navigational accuracy, and the human hippocampus"], "italic": ["Philosophical Transactions of the Royal Society B"], "year": ["1998"], "volume": ["353"], "issue": ["1373"], "fpage": ["1333"], "lpage": ["1340"]}, {"label": ["38"], "surname": ["Strang"], "given-names": ["G"], "italic": ["Linear Algebra and Its Applications"], "year": ["1988"], "publisher-loc": ["San Diego, Calif, USA"], "publisher-name": ["Harcourt Brace Jovanovich"]}, {"label": ["42"], "surname": ["Derdikman", "Fyhn", "Hafting", "Moser", "Moser"], "given-names": ["D", "M", "T", "MB", "EI"], "article-title": ["Breaking up the entorhinal grid in a hairpin maze"], "italic": ["Society for Neuroscience Abstract"], "year": ["2006"], "volume": ["33"], "fpage": ["68.10 pages"]}, {"label": ["46"], "surname": ["Heesy"], "given-names": ["CP"], "article-title": ["On the relationship between orbit orientation and binocular visual field overlap in mammals"], "italic": ["The Anatomical Record. Part A"], "year": ["2004"], "volume": ["281"], "issue": ["1"], "fpage": ["1104"], "lpage": ["1110"]}, {"label": ["48"], "surname": ["Hafting", "Fyhn", "Moser", "Moser"], "given-names": ["T", "M", "M-B", "EI"], "article-title": ["Phase precession and phase locking in entorhinal grid cells"], "italic": ["Society for Neuroscience Abstract"], "year": ["2006"], "volume": ["32"], "fpage": ["68.8 pages"]}, {"label": ["56"], "surname": ["Sharp", "Koester"], "given-names": ["PE", "K"], "article-title": ["Lesions of the mammillary body region severely disrupt the cortical head direction, but not place cell signal"], "italic": ["Hippocampus"]}]
Neural Plast. 2008 Jul 21; 2008:658323