the entorhinal cortex plays an   important 
 role in memory function .   in the rat , 

entorhinal cortex lesions impair performance in both spatial memory tasks   as well as in odor memory tasks . in monkeys ,
memory function 
 in delayed match to sample tasks is impaired by lesions of entorhinal cortex   and adjacent parahippocampal structures .
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 . 

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 . 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 [ 58 ] .  
pyramidal neurons in layer ii of medial 
 entorhinal cortex show persistent spiking that tends to turn on and off over 
 periods of many seconds .
pyramidal neurons in deep layers of entorhinal cortex can maintain 
 spiking at different graded frequencies for many minutes   as shown in figure 2(a ) .  
the 
 persistent spiking appears to due to muscarinic or metabotropic glutamate activation 
 of a calcium - sensitive nonspecific cation current [ 7 , 9 , 10 ] .  
this graded persistent 
 firing could allow these neurons to integrate synaptic input over extended 
 periods .  
entorhinal layer ii stellate cells show prominent subthreshold membrane 
 potential oscillations when depolarized near firing threshold [ 11 , 12 ] .  
these are small oscillations of a few 
 millivolts in amplitude that can influence the timing of action potentials   and can contribute to network oscillations [ 14 , 15 ] .  
the frequency of membrane potential 
 oscillations differs systematically along the dorsal to ventral axis of the 
 medial entorhinal cortex .
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 [ 1619 ] .  
the oscillations appear to 
 be due to a hyperpolarization activated cation current or h - current , that differs in time constant along the dorsal to ventral axis .   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 , but are observed in layer v pyramidal cells , where they may be caused 
 by m - current .  
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 [ 24 , 25 ] .  
examples of the pattern of firing in modeled 
 grid cells are shown in figures 1 and 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 [ 24 , 25 ] .
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 .  
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 [ 2630 ] .  
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 .
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 [ 3235 ] .  
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 figure 3(a ) .
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 v  =  [ dx / dt ,  dy / dt ] .  
integration of the velocity vector over a 
 period of time t after starting at 
 location x0 yields 
 the location vector x=  [ x ,  y ] at time t : 
 ( 1)x(t )  =  x0  +  0tv(t)dt . 
 
for example , if the velocity of an animal is 
 10 cm / sec in the x direction and 5 cm / sec in the y direction , integration over 
 5 seconds yields a final location of [ x , y ] = [ 50 cm , 25 cm ] relative to the start 
 location .
note that this integral 
 corresponds to the goal vector g(t )  =  x(t)x0 , 
 which is the negative of the return vector .
integration of the goal vector or return 
 vector could be provided by the mechanism of graded persistent firing in deep 
 layer entorhinal neurons [ 7 , 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   and in the postsubiculum , which provides direct input to the 
 entorhinal cortex [ 26 , 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   as well as in the hippocampus   and in the medial mammillary nucleus which receives output from the 
 postsubiculum and medial entorhinal cortex .   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 h.   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 h  =  hv. for example , a matrix representing two head 
 direction cells with preference angles 1 and 2 has two rows : h  =  [ cos  1sin  1cos  2sin  2 ] .   for a cell with a preference angle of   0
, the 
 activity of the head direction cell would be cos(0)dx / dt + sin(0)dy / dt = dx / dt .  
this framework has the advantage 
 that it allows for a simple inverse transform turning head direction space back 
 into cartesian coordinates .  
the inverse   of the matrix h is 
 ( 2)h1  =  [ sin2sin1cos2cos1]/(cos1sin2sin1cos  2 ) . 
 
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   and other regions [ 36 , 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 . 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 : 
 ( 3)a(t )  =  t=0th(v(t)dt )  =  h(x(t )    x(0 ) ) , 
 where 
 vector a(t ) represents the firing rate of a population of 
 graded persistent firing cells at time t.   for example , imagine two cells   a1 and a2 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 x direction for 4 seconds , and the scaling factor  is 0.25 hz / cm .  
moving in the x direction is 
 equivalent to moving at 0 degrees , which would result in the head direction 
 cells having the activity h1 = 1 and h2 = 0.5 . the 
 computation in ( 3 )
would then result in the frequency of the graded persistent 
 firing cells as follows : a  =  [cos  1sin  1cos  2sin  2][xy ]  =  0.25[100.50.87 ]  =  .  
thus , the graded persistent firing cells 
 would increase their activity to a1 =   10 hz and a2 = 5 hz .   mathematically , the inverse 
 transform of this firing rate vector computes the location vector x(t )  =  h1a(t)/ in cartesian coordinates ( see 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 .  
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 [ 39 , 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 .  
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   as well as the shift in firing location for grid cells between open 
 field scavenging and running on a long hairpin track   in which the view of each new segment could cause phase reset .
graded 
 persistent spiking could also hold initial head direction hd(0 ) and update this by integrating input from 
 neurons coding angular head velocity ahv(t ) in areas   such as the postsubiculum .
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 [ 24 , 43 , 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 figure 1(a ) ) .  
the second mechanism involves the graded 
 persistent firing shown in layer v ( see figure 2(a ) ) . in a general manner ,
the experimental data 
 on firing of single grid cells can be described by 
 ( 4)g(t )  =  cos(hx(t )  +   ) , 
 where g(t ) is the probability of 
 firing of the grid cell over time .  
the 
 product sign  represents 
 multiplication of the output from each row ( each head direction  ) of the head 
 direction transform matrix h described above .  
the description of the experimental data here 
 directly uses the vector representation of location over time x(t ) .  
orientation of the grid is determined by the 
 head directions  in h , the spatial phase is determined 
 by  , and the spacing between fields is determined by the angular 
 frequency .   this equation resembles other representations 
 of grid cells [ 18 , 45 ] but simplifies the 
 representation by using the head direction transform matrix .
the pattern of 
 periodic spatial firing of grid cells could arise from the pattern of cyclical 
 persistent firing as shown in 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 1(b ) and 1(c ) show   that the following equations   can generate grid cell firing properties:(5)h+dt  =  2v(t)(hv+(t))3/2,hdt  =  2v(t)(hv(t))3/2,vdt  =  h+(hv+(t))1/2h(hv(t))1/2,g(t )  =  [ v(t ) ] , 
  where h represents changes in current due to positive 
 components of the head direction input ( note that this uses cosine modulated 
 head direction input ) , and h 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 h transforming the rat movement velocity v(t ) .
the parameter  determines the time scaling of input effects 
 on activity levels . in the equations , v(t )
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 square brackets [ ] indicate 
 that spiking occurs whenever v(t ) crosses a threshold
. 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 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 .
examples of grid field plots obtained with this model are shown in 
 figures 1(b ) and 1(c ) .  
the grid fields 
 are more stable in the trajectory data from the moser laboratory than in 
 randomly created trajectories ( see figure 1(b ) ) or in a trajectory obtained in 
 our own laboratory ( see 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 . as an alternative to cyclical persistent firing
, graded persistent firing as shown in 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 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 .
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 .  
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 .
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 ) .
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 .    and o'keefe and burgess .
this mechanism uses the physiological fact 
 that depolarizing inputs to stellate cells cause a change in frequency of 
 membrane potential oscillations .  
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 : 
   ( 6)g(t )  =  [ (cos  t  +  cos(t  +  h0 t  h(v()d  +  x0 ) ) ) ] , 
  where g(t ) is the firing in time and space of a single modeled grid cell .  
 represents the baseline angular frequency of membrane 
 potential oscillations ( 2f ) in different 
 portions of the neuron .
h 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 h and the velocity vector v.   the inner product of each row of h with the 
 velocity vector v 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 x0 .  
this initial location vector is also transformed 
 by the matrix h. 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 ) .
this 
 model generates grid cell firing fields with spacing between fields dependent 
 upon the frequency of membrane potential oscillations [ 1618 ] .  
a grid cell created with 
 this model is shown in 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 .  
based on experimental data 
 alone , it appears that membrane potential oscillation frequency f is 
 scaled to grid cell spacing g by a constant factor f
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 . 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 : t=0th(v(t)dt  +  x0 )  =  hx(t ) .  
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 (t )  =  hx(t ) ( see figures 3(b ) and 3(c ) ) and the equation 
 for grid cell activity can be written for location as   g(t )  =  [ cos(hhx(t ) ) ] . 
  the location can be extracted from grid cell 
 phase by using the inverse of the head direction transform matrix as follows : x(t )  =  h1(t)/. figure 3(d ) shows the trajectory obtained from 
 this inverse transform of phase . 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 , a phenomenon known as theta phase precession .  
the phenomenon of theta phase precession was 
 initially shown for place cell firing in the hippocampal formation [ 49 , 50 ] and was proposed to 
 arise from the interaction of network theta rhythm oscillations with cellular 
 theta rhythm oscillations [ 49 , 51 , 52 ] .  
as an alternative model , precession was 
 proposed to arise from the readout of sequences of place cell activity [ 5355 ] .
the oscillatory interference model of grid 
 cells [ 18 , 47 ] extended the models of 
 hippocampal phase precession and can account for grid cell phase precession .   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 d  s , and ( 2 ) the pattern of phase precession depends on a higher 
 frequency component that corresponds to the mean of the two frequencies ( s + d)/2 .
the phase of this 
 second - high frequency component of the summed oscillation is sum  =  t  +  0 t  hv()d/2 .  
the spiking will occur near the peak phase of 
 this summed oscillation which is equal to some multiple n of the full 
 cycle : sum = n2. 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   x  =  0thv()d.   note also that the phase of the soma 
 oscillations is the product of soma frequency and time : soma = 2ft .   therefore , the equation for the phase of the 
 summed oscillation can be reduced to sum = soma + bx/2 .
plotting of theta phase precession 
 essentially involves plotting the timing of spikes ( which occur when sum = n2 ) 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  ) .  
thus , the vertical axis of a plot of theta 
 phase precession shows the phase of the soma oscillation at the time of each 
 spike : soma = n2   f bx plotted relative to location x on the 
 horizontal axis .  
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   but only covers about 180 degrees of the network theta oscillation 
 cycle .
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 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 ) .
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 h. 
 if oscillation phases are reset to zero at the starting location , the 
 return vector r 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 r=h1/.   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 ) . path 
 integration
based on idiothetic cues alone builds up substantial error   that is usually corrected by comparison with sensory cues such as 
 visual stimuli .  
they maintain the same properties when returned to a familiar room , and they change their scale during a period of time after 
 manipulations of environment size .  
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 .  
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 .
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 .  
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 [ 27 , 56 ] as well as in structures 
 including the anterior dorsal thalamic nucleus .  
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 .
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 .
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 .  
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 [ 16 , 57 ] , this could allow grid 
 cells to provide an effective mechanism for representations of state and action 
 in continuous space and time .
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 . as shown in figure 6
, this 
 model uses a functional loop that encodes and retrieves trajectories via three 
 stages : ( 1 ) 
 head direction cells h(t ) update grid cells , ( 2 ) grid cells g(t ) update place cells , and ( 3 ) place cells p(t ) activate associated head direction 
 activity .  
this model is consistent 
 with the anatomical connectivity ( see 
 figure 6 ) .  
the 
 head direction cells could update grid cells via projections from the 
 postsubiculum ( dorsal presubiculum ) to the medial entorhinal cortex [ 5860 ] , causing updating of persistent firing as described above , or 
 influencing the phase of membrane potential oscillations [ 1618 ] .  
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 [ 45 , 61 ] .  
the connectivity from grid cells to place 
 cells could be formed by different computational mechanisms [ 45 , 62 , 63 ] .  
place cells can become associated with head 
 direction activity via direct projections from region ca1 to the postsubiculum , or via indirect projections from region ca1 to the subiculum [ 61 , 64 ] , and projections from 
 the dorsal and distal regions of the subiculum to the postsubiculum and medial 
 entorhinal cortex , both of which contain head direction cells . 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 whp with strengthened 
 connections between active place cells p(t ) and the head direction cell 
 activity vector h(t ) as follows : in this equation , head direction vectors 
 associated with individual place cell locations are identified with the place 
 cell index p.   during retrieval , 
 the head direction activity depends upon synaptic input from place cell 
 representations as follows : h(t ) = whp p(t ) .
this model has the capacity for performing 
 episodic encoding and retrieval of trajectories in simulations , including trajectories based on experimental data or trajectories 
 created by an algorithm replicating foraging movements of a rat in an open 
 field .   during encoding ,
a series 
 of place cells p(t ) is 
 created associated with particular locations xp = x(t ) .   each place cell is also associated with input 
 from the grid cell population activity g(t ) and with the head direction vector    hp = h(t ) that occurred during the initial movement from 
 that location .   for retrieval , the 
 simulation is cued with the grid cell phase vector (t0 ) and head direction vector h(t0 ) present at the start location .  
the head 
 direction vector updates the grid cell phase vector d(t ) , 
 which alters the activity of grid cells . 

the grid cell firing drives place cells p associated with 
 subsequent locations on the trajectory . 

the activation of each new place cell 
 activates a new head direction vector hp 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   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 . 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 .  
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 .  
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 .  
the retrieval mechanism 
 mediated by place cells can be enhanced by inclusion of cells that fire 
 dependent upon the arc length of the trajectory , or by the time interval alone .  
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 can not 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   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 .
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
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 .
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 .
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 .
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 .
 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 . 
1. INTRODUCTION
2. PHYSIOLOGICAL DATA
3. PATH INTEGRATION
4. INTEGRATION BY PERSISTENT FIRING CAN CODE LOCATION
5. PERSISTENT SPIKING COULD UNDERLIE GRID CELL FIRING
6. INTEGRATION BY MEMBRANE POTENTIAL OSCILLATION PHASE CAN CODE LOCATION
7. MODEL OF EPISODIC MEMORY
8. CONCLUSIONS