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Mathematical Modeling of Hippocampal Spatial Memory with Place Cells

Module by: Georgene Jalbuena. E-mail the author

Summary: This report summarizes work done as part of the Modeling Spatial Memory with Place Cells PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem. This work was studied in the Rice University VIGRE program in the Summer of 2011. In this module, we mathematically model Kneirim's Double Rotation Experiment with a model given by Cox and Gabbiani and analytically discuss the relation between overlapping place fields and synaptic input weights.

Background: Place Cells in the Hippocampus

Biology

Spatial memory is what allows us to keep track of our location in space by making mental maps of each environment. Let's consider what happens in the brain during the process of forming these internal maps.

Connections, called synapses, between certain neurons strengthen or weaken–a process known as synaptic plasticity. The strength, or weight, of a synapse controls how much one neuron can affect another. Synaptic plasticity is necessary for memory formation [4].

While many neurons are involved with spatial memory, our focus is upon neurons in the hippocampus called place cells [4]. Place cells have a unique firing pattern. When an environment becomes familiar, each place cell becomes associated with one area of the environment. In other words, after repeated exposure to one environment, a place cell will come to spike in only one area of the environment, which is called that cell's place field [4]. See Figure 1.

Figure 1: The place field of one place cell. As a rat explored the square environment, experimenters monitored the activity of one place cell.ÊThe black line is the rat's trajectory, and each red dot represents a location that the place cell has become active. The place field of the particular place cell that was monitored is the area of the environment marked densely by red dots [4].
Figure 1 (Moser-Placefield.png)

Due to place cells' characteristic firing pattern where a cell has a single place field in each environment, it is easy to test if a rat recognizes the environment they have been placed in by examining place cell activity.

Motivation: Double Rotation Experiment

Our research on place cells is based off of the Double Rotation Experiment (DRE) conducted by our collaborator Dr. Knierim [2].

Figure 2: The Double Rotation Experiment (DRE). Rats were trained to walk clockwise around the track, and the place fields corresponding to specific locations on the track were monitored. During the learning phase, the place fields experienced a backward shift. During double rotation runs of the experiment, the local cues were rotated backward (counterclockwise), and the place fields shifted backward as well. Thus the place fields seemed to favor following the local cues. Perhaps the natural backward shift seen in the learning phase predisposes the place fields to wanting to follow cues that move backwards [2].
Figure 2 (doublerotationReal.png)

His team set up a track that a rat traversed and monitored the place fields that corresponded to specific locations on the track. Dr. Knierim and his team set up local cues on a circular path and distal cues on surrounding curtains and trained rats to walk along the path in a clockwise direction. The place fields corresponding to specific locations on the track were monitored. See Figure 2.

During the learning phase, as the rat completed laps and became more familiar with the layout, the place fields shifted backward along the path opposite the rat's movement. In standard runs of the experiment, the rat ran clockwise around the path with the layout that it learned. In double rotation runs of the experiment, the local cues were rotated counterclockwise (backward) and distal cues were rotated in clockwise–hence the “double rotation". During the standard and the double rotation runs, the spatial inputs and place fields did not always match. In the double rotation runs, the spatial inputs tended to follow the distal cues, whereas the place fields tended to follow local cues.

One thing that might explain this is the backward shift seen in the learning phase. The natural backward shift of place fields that happens when a rat has become familiar with its environment appears to bias the place fields to follow the cues that shift backwards. The backward shift of place fields has been observed in other experiments as well when rats became familiar with a path [3]. It may be that the backward shift is a part of the learning process. Thus, we want to understand more about the backward shift and aim our research in this direction.

The backward shift may be dependent upon the weights of synapses, yet the dependence is not well understood. We focus on how the dynamic weights due to synaptic plasticity affect the backward shift of place fields.

We implemented a far simplified model of the DRE to try to understand this relationship better.

Modeling

Here we will describe the models used throughout the summer. A circuit model allows us to understand the mechanics of one neuron, while a 120-cell ring models the Double Rotation Experiment [2], and a single-cell model was used for the analysis of weights and backward shift. Refer to Table 1 for the parameters used throughout the models and their values.

Table 1: Parameters for models. These values are used unless otherwise specified.
Parameter Value Description
w i n p w i n p 2winp202winp20 mV Input weight
I I 2I302I30 ms Interspike interval of external inputs
τ τ 20 ms Decay time
d t d t 1 ms Time step
v r v r -70 mV Resting voltage
v t h v t h -54 mV Threshold voltage

Neuron as a Circuit: Integrate and Fire

We model a single place cell with the Integrate and Fire (IAF) model so that we can accurately calculate the subthreshold voltage and approximate spike times of a cell at little computational expense [5].

Figure 3: Circuit model of a cell. The cell can be modeled as a circuit with three currents. The difference between potentials φinφin and φoutφout gives the reversal potential vClvCl, which we refer to as vv in the text. gClgCl and gsyngsyn denote the conductances of the chloride channels and the synapse, respectively. CmCm is the membrane capacitance. vsynvsyn depends upon the equilibrium concentrations of the ion of the associated channel. By Kirchoff's current law, ICl+IC+Isyn=0ICl+IC+Isyn=0, [1].
Figure 3 (Circuit.png)

The cell can be modeled as a leaky capacitor that separates charge by controlling the flow of ions across the cell membrane, making a difference between potentials φinφin and φoutφout across the membrane (v=φin-φoutv=φin-φout). This model approximates the subthreshold voltage (vv before it reaches the threshold voltage vthvth) and the time that vv does reach threshold vthvth. When vv reaches the threshold voltage, vv experiences a sharp increase then decrease and the cell is said to have “spiked" or “fired".

Let CmCm denote the membrane capacitance, gClgCl and gsyngsyn denote the conductances of the chloride channels and the synapse, respectively, and vClvCl denote the reversal potential (the voltage at which no net flow of chloride ions occurs). vsynvsyn is determined by the equilibrium concentrations of the ion of the associated channel, [1]. Input from other cells adds excitatory synaptic current. This input allows for the cell to depolarize and eventually reach vthvth and fire, which sends an electric signal to neighboring cells. The electric signal from a neighboring cell allows for voltage-gated channels to open, which affects the synaptic conductance gsyngsyn. We set the reversal potential above a threshold voltage so that as gsyngsyn increases and the channels open, the cell's voltage vv approaches the threshold vthvth. When vvthvvth, the cell fires [1]. See Figure 3 for a model of a cell as a circuit [1]. The synaptic conductance is governed by the ODE

τ g s y n ' = - g s y n + i w i i n p n δ ( t - T n ) , τ g s y n ' = - g s y n + i w i i n p n δ ( t - T n ) ,
(1)

where ττ is the decay constant, wiinpwiinp is the weight of synaptic input from the iith synapse, TnTn is the set of input spike times for the presynaptic cell ii, and δδ is the Dirac delta function. From Dr. Cox's book [1], we see that applying Kirchoff's current law results in

C m d v d t + g C l ( v - v C l ) + g s y n ( v - v s y n ) = 0 . C m d v d t + g C l ( v - v C l ) + g s y n ( v - v s y n ) = 0 .
(2)
Figure 4: Simple 2-cell network. Cell 1 receives input from an external source, and Cell 2 receives input from Cell 1. The conductances and voltages of the two cells are calculated by equations Equation 1 and Equation 2, respectively [1].
Figure 4 (model2cell.png)

Model of DRE: 120-cell Ring

Figure 5: 120-cell ring. We put 120 cells, represented by circles, in a ring architecture with bidirectional interactions between cells, as well as synapses to sources of external input.ÊLike the DRE, each place cell has a corresponding place field on a circular path that a simulated rat traverses clockwise. The bidirectional interaction between two neighboring cells is modeled through two synapses with one synapse for each direction [1]. The weights of the synapses changes over time according to the STDP model described in "STDP"[6].
Figure 5 (ring120cell.png)

We use the IAF model for single cells, and we connect these cells into a ring of 120 place cells to simulate the DRE [1]. The 120-cell ring is depicted in Figure 5. Each cell receives external spatial input as well as input from neighboring cells. The conductances and voltages of Cells 1 and 2 are depicted in Figure 6 as calculated by equations Equation 1 and Equation 2. We monitor the weights of the connections between neighboring cells over time, where there is an arbitrary maximum weight bound so that the weights do not approach infinity and a minimum weight bound of 0 so the weights do not become negative. We also monitor how the changes of the weights affect the position of the place fields. See Figure 6 for a depiction of the IAF model.

Figure 6: Depiction of the IAF model for Cell 1 and Cell 2 from Figure 5. We calculate the conductances and voltages for Cells 1 and 2 by Equation 1 and Equation 2, respectively. vth=-54mVvth=-54mV, and wext=w12=2Swext=w12=2S are the weights of the synapses from the external source and Cell 1, respectively. It can be seen that in Place Field 1 (0<t<1000<t<100), Cell 1 receives external input every 20 ms (in the form of conductance spikes), and in Place Field 2 (100t<200100t<200), Cell 2 receives the external input. At each external input spike, the corresponding cell fires (its voltage spikes) and gives a small amount of input to its neighboring cells. (IAF2cellsweight.m)
Figure 6 (IAF2cells.pdf)

STDP

To determine how the weights of the synapses change, we use an STDP model [6]. The spikes times of the pre- and post-synaptic cells are compared, and the smaller the time difference, the more the weight is adjusted. See Figure 7. The percentage of weight change is determined by

F ( Δ t ) = A + e Δ t / τ + , Δ t < 0 - A - e - Δ t / τ - , Δ t 0 , F ( Δ t ) = A + e Δ t / τ + , Δ t < 0 - A - e - Δ t / τ - , Δ t 0 ,
(3)

where Δt=tpre-tpostΔt=tpre-tpost and A+A+ and A-A- scale the maximal amount of change allowed when ΔtΔt is close to 0 [6]. F(Δt)F(Δt) is depicted in Figure 7.

Figure 7: Percentage of weight change due to STDP. This graph shows that as Δt=tpre-tpostΔt=tpre-tpost approaches 0, the percentage of change in the weight of the synapse increases [6]. F(Δt)F(Δt) is calculated by Equation Equation 3.
Figure 7 (STDPweightchange.png)

In our 120-cell model, we set a lower bound for the weights at 0mV0mV and an upper bound at 5mV5mV. The necessity for an upper weight bound is one of the weaknesses of the STDP model, so Andrew Wu, another member of this PFUG, has done work with other plasticity models. See the link "Mathematical Models of Hippocampal Spatial Memory".

The work with the 120-cell model is all computational. Our results show that each lap, the weight of the synapse from Cell 1 to Cell 2 (w12w12) increases toward a set weight bound and the weight of the synapse from Cell 2 to Cell 1 (w21w21) decreases to 0 as in Figure 8. We also see that after 4 laps around the path, the place fields start to shift backward. Figure 9 shows that the spike time of Cell 2 decreases each lap, which is indicative of a backward shift of Cell 2's place field. See IAF120cells1stspks.m.

Figure 8: Weights of synapses over time. The weight of the synapse from Cell 1 to Cell 2 (w12w12) increases each lap (1 lap = 12,000 ms) as the weight of the synapse from Cell 2 to Cell 1 (w21w21) decreases each lap to 0. (IAF120cellsSTDP.m)
Figure 8 (STDPweights.pdf)
Figure 9: Backward shift of place field of Cell 2 in msms. The rat spends 100ms100ms in each place field. It can be seen here that after as few as 4 laps around the track, place cell 2 starts to fire earlier and its place field shifts backwards. (IAF120cells1stspks.m)
Figure 9 (IAF120cells1stspks.pdf)

Model for Analysis

Figure 10: Simple 1-cell system. Cell 1 receives input of weight winpwinp at an interspike interval II.
Figure 10 (model1cell.png)

We begin by considering only one place cell which receives input from one external source with constant weight winpwinp at a set interspike interval, II, as depicted in Figure 10. The following equation gives the voltage vv in mVmV of the cell at time tt with nn total input spikes:

τ v ' ( t ) = ( v r - v ( t ) ) + w i n p i = 1 n δ ( t - T i ) τ v ' ( t ) = ( v r - v ( t ) ) + w i n p i = 1 n δ ( t - T i )
(4)

where τ=20τ=20msms is the membrane time constant, vr=-70vr=-70mVmV is the resting potential, TT is the set of input spike times, and δ(t-Ti)δ(t-Ti) is the Dirac delta function. This is a simplification of equation Equation 2.

Computational Method

To solve for v(t)v(t) computationally, we first look at the times with no input spikes (kI<t<(k+1)IkI<t<(k+1)I). Integrating both sides of equation Equation 4 from t-dtt-dt to tt and using the trapezoid rule, we find

τ ( v ( t ) - v ( t - d t ) ) = v r d t - v ( t ) + v ( t - d t ) 2 , which can be rearranged as v ( t ) = 2 d t 2 τ + 1 · v r + 2 τ - 1 2 τ + 1 · v ( t - d t ) . τ ( v ( t ) - v ( t - d t ) ) = v r d t - v ( t ) + v ( t - d t ) 2 , which can be rearranged as v ( t ) = 2 d t 2 τ + 1 · v r + 2 τ - 1 2 τ + 1 · v ( t - d t ) .
(5)

When there is an input spike, we add winpwinp to v(t)v(t), which is shown in

v i n p ( t ) = v ( t ) + w i n p . v i n p ( t ) = v ( t ) + w i n p .
(6)

Analytic Method

To solve for v(t)v(t) analytically, we first look at v(t)v(t) between input spikes. From equation Equation 4, we get

τ v ' ( t ) = ( v r - v ( t ) ) . τ v ' ( t ) = ( v r - v ( t ) ) .
(7)

Solving this ordinary differential equation gives us

v ( t ) = v r + c e - t / τ , v ( t ) = v r + c e - t / τ ,
(8)

where cc is the constant of integration. We know we want v(0)=vr+winpv(0)=vr+winp, so cc must equal winpwinp. Thus, we have

v ( t ) = v r + w i n p e - t / τ , where 0 t < I , v ( t ) = v r + w i n p e - t / τ , where 0 t < I ,
(9)

which simply tells us that after one input spike at t=0t=0, winpwinp decays so that v(t)v(t) approaches vrvr. Consider the following calculations of v(t)v(t) for up to three input spikes.

At t=It=I, we have a second input spike, and at I<t<2II<t<2I, we decay the input to find

v ( I t < 2 I ) = v r + w i n p e - t / τ + w i n p e - ( t - I ) / τ . v ( I t < 2 I ) = v r + w i n p e - t / τ + w i n p e - ( t - I ) / τ .
(10)

Finally, at t=2It=2I, we have a third input spike and see

v ( 2 I ) = v r + w i n p e - 2 I / τ + w i n p e - I / τ + w i n p . v ( 2 I ) = v r + w i n p e - 2 I / τ + w i n p e - I / τ + w i n p .
(11)

To determine when the voltage reaches threshold and the cell spikes, we need only examine the peak values of vv, which are when t=kI,0kn-1t=kI,0kn-1. Thus, we use the following generalized formula to calculate v((n-1)I)v((n-1)I) when there are nn total input spikes:

n , v ( ( n - 1 ) I ) = v r + w i n p k = 0 n - 1 e - I / τ k . n , v ( ( n - 1 ) I ) = v r + w i n p k = 0 n - 1 e - I / τ k .
(12)

Figure 11 shows that in the absence of spikes, the peak voltages approach an asymptote. This asymptote can be calculated by

v = lim n v ( ( n - 1 ) I ) = v r + w i n p k = 0 e - I / τ k = v r + w i n p 1 1 - e - I / τ . v = lim n v ( ( n - 1 ) I ) = v r + w i n p k = 0 e - I / τ k = v r + w i n p 1 1 - e - I / τ .
(13)

If v<vthv<vth, then the cell will never spike.

Figure 11: Voltage as a function of time as calculated by equation Equation 12. The peak voltages are denoted by asterisks. Here we set vth=-52mVvth=-52mV. (AnpeakV.m)
Figure 11 (AnpeakV.pdf)

Problems and Results

Minimum input weight for activity

Computational vs. Analytic Method

We found the minimum input weight winpwinp necessary for the cell to spike at least once as a function of the input time interval II when given a sufficiently long simulation.

Let the interspike interval II and input weights winpwinp satisfy 2I302I30 and 2winp202winp20.

In the computational method, the Matlab program compW.m calculates v(t)v(t) according to equations Equation 5 and Equation 6. In AnalysisW.m, the minimum winpwinp is calculated by

w i n p = ( v t h - v r ) ( 1 - e - I / τ ) , w i n p = ( v t h - v r ) ( 1 - e - I / τ ) ,
(14)

which was obtained by setting vv of equation Equation 13 to vvthvvth where

v t h v r + w i n p 1 1 - e - I / τ . v t h v r + w i n p 1 1 - e - I / τ .
(15)

Figure 12 shows that as the input time interval increases, greater input weight is necessary for the cell to spike at least once (AnalysisW.m). We note on the graph the value of winp=10.11winp=10.11 at I=20I=20 because these two values will be put to use in the next section.

Figure 12: Comparison of winpwinp from computation and analysis as a function of II. v(t)v(t) is calculated by equations Equation 5 and Equation 6 in compW.m. winpwinp is calculated by equation Equation 14 in AnalysisW.m. (Plotted in AnalysisW.m)
Figure 12 (BothWinpsxI.pdf)

Number of input spikes versus input weight

Computational vs. Analytic Method

We determine the minimum number of input spikes necessary for the cell to spike as a function of input weight.

We use I=20I=20 and consider only the weights that produce at least one spike with sufficient simulation, starting with winp=10.2winp=10.2 as shown in Figure 12. Let n1n1 denote the minimum number of input spikes of weight winpwinp necessary for v(t)v(t) to reach vthvth. We see that

n 1 - τ I · ln 1 - v t h - v r w i n p · 1 - e - I / τ . n 1 - τ I · ln 1 - v t h - v r w i n p · 1 - e - I / τ .
(16)

In the computational method, the Matlab program compT.m calculates n1n1 by updating v(t)v(t) with equations Equation 5 and Equation 6. In the analytic method, AnalysisT.m calculates n1n1 with equation Equation 16.

Figure 13 shows that n1n1 is clearly a step-wise function where higher input weights allow the cell to spike after fewer input spikes.

Figure 13: Comparison of minimum number of input spikes necessary for activity from computation and analysis as a function of winpwinp. v(t)v(t) is calculated by equations Equation 5 and Equation 6 in compT.m and by equation Equation 12 in AnalysisT.m. (Plotted in AnalysisT.m)
Figure 13 (BothNinpxWinp.pdf)

Application of findings

Figure 14: Simple 2-cell network with synapses active in Place Field 1. While the rat is in the place field of Cell 1, Cell 1 receives input with weight wextwext from an external source, and when Cell 1 spikes, it gives input with weight w12w12 to Cell 2.
Figure 14 (PF1-2cells.png)

We apply the results from the two previous sections to solve a couple simple questions. Consider the situation depicted in Figure 14: the rat is in Place Field 1, Cell 1 receives input with weight wextwext from an external source, and, when Cell 1 spikes, it gives internal input to Cell 2 with weight w12w12. We can find the minimum weight of w12w12 necessary for Cell 2 to spike in Place Field 1, or where Place Field 1 and Place Field 2 overlap.

For a fixed interspike interval II of external input of fixed weight wextwext, we calculate n1n1 using equation Equation 16 as a function of II and wextwext. Thus n1n1 denotes the minimum number of external input spikes of weight wextwext necessary for Cell 1 to fire. Let us denote the time of Cell 1's first spike as

t 1 = ( n 1 - 1 ) I . t 1 = ( n 1 - 1 ) I .
(17)

While the rat is in Place Field 1, Cell 2 only receives input from Cell 1. To find the interspike interval of Cell 1's spikes, or equivalently the interspike interval that Cell 2 receives input in Place Field 1, consider Figure 15. We know that Cell 1's first spike is at t=t1t=t1. After it spikes, the voltage decays until the next input spike at n1In1I. Then it takes another (n1-1)Ims(n1-1)Ims for Cell 1 to spike at t=n1I+(n1-1)It=n1I+(n1-1)I. Thus we subtract the first spike time from the second spike time

n 1 I + ( n 1 - 1 ) I - ( n 1 - 1 ) I = n 1 I , n 1 I + ( n 1 - 1 ) I - ( n 1 - 1 ) I = n 1 I ,
(18)

and we see that after the first spike time at t1=(n1-1)Imst1=(n1-1)Ims, Cell 1 fires every n1Imsn1Ims.

Figure 15: Interspike interval of Cell 1 from external input alone, I1I1. Cell 1's first spike occurs at t=t1=(n1-1)It=t1=(n1-1)I. Then the voltage of Cell 1 decays until the next input spike at t=n1It=n1I. It then takes another (n1-1)Ims(n1-1)Ims for Cell 1 to spike at t=n1I+(n1-1)It=n1I+(n1-1)I. Thus, it can be seen that except for the first spike at t=(n1-1)It=(n1-1)I, Cell 1 fires every n1Imsn1Ims.
Figure 15 (cell1spkfreq.png)

Thus, we know that Cell 1 gives input to Cell 2 with weight w12w12 every n1Imsn1Ims. To find the minimum weight of w12w12 that would allow for Cell 2 to fire in Place Field 1, we consider two cases where the first is simpler: Place Field 1 is infinitely long or finitely long.

Equation Equation 14 tells us that if

w 12 ( v t h - v r ) ( 1 - e - n 1 I / τ ) , w 12 ( v t h - v r ) ( 1 - e - n 1 I / τ ) ,
(19)

then Cell 2 would fire given a sufficiently long Place Field 1. Thus, if Place Field 1 is infinitely long, we simply require that equation Equation 19 be true, then Cell 2 fires in Place Field 1. As we do in "Number of input spikes versus input weight", we choose an interspike interval: let n1I=40msn1I=40ms. For this value of n1In1I, App.m calculates the minimum of equation Equation 19 to be 13.83mV13.83mV, as marked in Figure 16.

Figure 16: Minimum w12w12 necessary for Cell 2 to fire as a function of n1In1I. It is clear that the larger the interspike interval, the more weight is required for activity. We note the value of w12w12 at n1I=40n1I=40 because these values are used in the case where Place Field 1 is finite. (App.m)
Figure 16 (w12xn1I.pdf)

Now assume Place Field 1 is finitely long. We apply equation Equation 16 to find n2n2, where n2n2 is the minimum number of input spikes of weight w12w12 necessary for Cell 2 to fire. We let n1I=40msn1I=40ms and consider only the values of w1213.83mVw1213.83mV as calculated for n1I=40mVn1I=40mV in equation Equation 19. Figure 17 shows n2n2 as a function of these values of w12w12. As in "Number of input spikes versus input weight", n2n2 is step-wise like n1n1.

Figure 17: Minimum number of input spikes necessary for Cell 2 to fire, denoted as n2n2, as a function of input weight w12w12. We choose n1I=40msn1I=40ms and consider only the values of w1213.83mVw1213.83mV (the minimum w12w12 necessary for Cell 1 to fire from equation Equation 19). Like n1n1 from "Number of input spikes versus input weight", n2n2 is step-wise. We note the minimum weight necessary for Cell 2 to fire in Place Field 1, w12=16.01mVw12=16.01mV. (App.m)
Figure 17 (n2xw12.pdf)

The first spike of Cell 2 occurs at time t2=(n2)(n1I)-It2=(n2)(n1I)-I. We find the value of w12w12 necessary for t2t2 the time spent in Place Field 1. Suppose the time spent in Place Field 1 is 50ms50ms. We see from Figure 17 that

n 2 = 2 if 13 . 83 w 12 15 . 95 1 if 15 . 95 < w 12 20 . 00 n 2 = 2 if 13 . 83 w 12 15 . 95 1 if 15 . 95 < w 12 20 . 00
(20)

and calculate that

t 2 = 60 if 13 . 83 w 12 15 . 95 20 if 15 . 95 < w 12 20 . 00 t 2 = 60 if 13 . 83 w 12 15 . 95 20 if 15 . 95 < w 12 20 . 00
(21)

Thus, for 13.83w1215.95mV13.83w1215.95mV, t2=60mst2=60ms and Cell 2 will not fire in Place Field 1, but for 16.01w1220.00mV16.01w1220.00mV, t2=20mst2=20ms and Cell 2 will fire in Place Field 1. The minimum value of w12w12 for t250mst250ms is marked on Figure 17 (App.m).

Future work

Our goal is to better understand the relation between input weights and backward shift of place fields. We modeled the Double Rotation Experiment using a simple 120-cell ring and calculated the backward shift of the place fields as a function of input weights of that model [2], [1]. We have computationally and analytically found the minimum input weight necessary for activity as a function of the interspike interval as well as the minimum number of input spikes necessary for activity as a function of input weight. We applied our findings analytically to find the minimum internal input weight necessary for place fields to overlap in a simple 2-cell model.

Future work may include constructing a code to compare our analytical results from "Application of findings" for the infinite and finite place field cases that could give us the minimum internal input weight necessary for place fields to overlap. We could also couple the equations regarding input weight changes by spike timing-dependent plasticity with the equation that gives us the minimum number of input spikes necessary for activity that is dependent upon input weights. Since the time of the first spike of a cell can be given in terms of the minimum number of input spikes necessary for activity as a function of input weight, we may be able to find a value that the time approaches. Ultimately, given a set maximum for the input weight, we would like to be able to predict the amount of backward shift of a place field using spike timing-dependent plasticity.

References

  1. Gabbiani, F. and Cox, S. J. (2010). Mathematics for Neuroscientists. Elsevier Academic Press.
  2. Lee, I. and Yoganarasimha, D. and Rao, G. and Knierim, J. J. (2004, July). Comparison of population coherence of place cells in hippocampal subfields CA1 and CA3. Nature, 430, 456-459.
  3. Mehta, M. R. and Barnes, C. A. and McNaughton, B. L. (1997, August). Experience-Dependent, Asymmetric Expansion of Hippocampal Place Fields. Proceedings of the National Academy of Sciences of the United States of America, 94(16), 8918-8921.
  4. Moser, E. I. and Kropff, E. and Moser, M. B. (2008, February). Place Cells, Grid Cells, and the Brain's Spatial Representation System. Annual Review of Neuroscience, 31, 69-89.
  5. Ward, Kathryn Ruth. (2009, April). The Effect of Synaptic Plasticity on Spatial Representation and Navigation. Masters thesis. Rice University.
  6. Yu, X. and Knierim, J. J. and Lee, I. and Shouval, H. Z. (2006, February). Simulating place field dynamics using spike timing-dependent plasticity. Neurocomputing, 69, 1253-1259.

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Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

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My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks