Scalar timing in memory: A temporal map in the hippocampus

Abstract

Decision making, rate calculation, and planning are examples of daily tasks we perform. They require accurate timing and involve many cortical areas such as the prefrontal cortex, the striatum, and the hippocampus. Although the neurobiological origin and the mechanisms of interval timing are largely unknown, we developed increasingly accurate mathematical and computational models that can mimic some properties of time perception. The accepted paradigm of temporal durations storage is that the objective elapsed time from the short-term memory is transferred to the reference memory using a multiplicative “memory translation constant” K*. It is believed that K* has a Gaussian distribution due to trial-related variabilities. To understand K* genesis,we hypothesized that the storage of temporal memories follows a topological map in the hippocampus, with longer durations stored towards dorsal hippocampus and shorter durations stored toward ventral hippocampus. We found that selective removal of memory cells in this topological map model shifts the peak-interval time in a manner consistent with the current experimental data on the effect of hippocampal lesions on time perception, and opens new avenues for experimental testing of our topological map hypothesis. We found numerically that the relative shift is determined both by the lesion size and its location and we suggested a theoretical estimate for the memory translation constant K*.

1. Introduction

The perception and use of durations in the supra-second range (interval timing) is essential for survival and adaptation, and is critical for fundamental cognitive processes like decision making, rate calculation, and planning of actions [1]. In the vast majority of species, protocols, and manipulations to date, interval timing is time-scale invariant: time-estimation errors increase linearly with the estimated duration [2, 3, 4, 5, 6] (Fig. 1). Time-scale invariance is ubiquitous in many species [1, 3] from invertebrates to fish, birds, and mammals, such as mice [7], rats [8], and humans [9]. Scale invariance is the fundamental property of interval timing, as it is extremely stable over behavioral (Fig. 1), lesion [10], pharmacological [11, 12, 13], and neurophysiological manipulations [14, 15].

Figure 1: Time-scale invariant interval timing.

The time-in-nosepoke (TIN) curves for mice timing a 20 s interval (A) or 40 second interval (C) overlap when normalized by the maximum TIN (vertical axis), and respectively, by the criterion time (horizontal axis); redrawn from [7].

One of the most influential theoretical explanation for time perception has been the Scalar Expectancy Theory (SET) [5] further augmented with the information-processing theory in the seminal work of Church [16], Gibbon and Church [17], and Gibbon, Church, and Meck [6]. Without further detailing the SET framework, we only mention its key elements: a clock process consisting of a pacemaker and an accumulator, a memory process consisting of short-term and reference memory stores, and a comparator process where decisions are made that lead to behavioral output [5, 6, 16, 17]. The reference memory in SET theory serves two important purposes: (1) provides a temporal reference by storing “important” times, such as the reinforcement time, and (2) holds the key for the observed scalar property of timing, i.e. the standard deviation is proportional to the mean estimated time [18].

In SET theory, a multiplicative transformation, the famous “memory translation constant” K*, was introduced to mediate between the short-term (working) and long-term (reference) memory [6, 16, 17, 19]. In other words, the difference between the actual duration (clock reading) and the encoded duration stored in the reference memory is determined by the memory translation constant K* (see [20] for a detailed review of the SET developed by Gibbon [5]). It is assumed that the pacemaker/accumulator system faithfully represent the objective elapsed time in terms of the number of pacemaker pulses. At the time the content of the accumulator, i.e. the short-term memory, is transferred to the reference memory its content is multiplied by a “memory translation constant” K*. K* is not really a constant but was rather assumed to have a Gaussian distribution, presumably produced because of the accumulation of a large number of presentations of reinforcers [6, 16, 17, 18]. Arguably the two most influential theoretical studies that attempted a mathematical perspective on how the Gaussian K* produced individual trial responses are Gibbon et al. [21] and Brunner et al. [22]. In this study, we further investigated the nature of the “memory translation constant” K* starting from two assumptions: (1) a Gaussian representation of the reinforcement time in the reference memory (see [6, 16, 17, 21, 22]), and (2) possible peak-interval shifts correlated with the spatial location of hippocampus lesions (see [23, 24, 25, 26, 27]). To bridge the above two paradigms, we suggested here a mathematical model of a topological organization of hippocampus and numerically investigated the effect of lesions in such a model.

Hippocampal lesions and interval timing.

Hippocampal lesions have been shown to affect peak time in peak-interval procedures and the subjective equivalence points in temporal bisection procedures [10, 28, 29, 30]. Rats with hippocampal damage responded earlier than the scheduled time of reinforcement in a variety of peak-interval procedures [28, 31]. Hippocampal lesions also disrupted responses in differential reinforcement of low rates (DRL) schedules. In DRL, rats are trained to withhold responding for food until after a set time has elapsed (e.g. more than 15 s). Rats with dorsal, ventral, or complete hippocampal lesions are highly inefficient at this task because they significantly diminish rats’ ablility to wait for the set temporal interval to elapse [32]. Consequently, it has been suggested that the hippocampus plays an important role in temporal memory and/or inhibitory processes [33].

Importantly, both pre-training and post-training dorsal hippocampal (DH) lesions produced leftward shifts in peak times, confirming previous investigations and suggesting a possible role for the DH in the cortical striatal-based timing mechanisms [25, 26, 28, 30, 31, 34]. In contrast, ventral hippocampal (VH) lesions produced a temporary rightward shift of peak times [27]. Moreover, when peak times and peak rates were modulated by reversal learning, pre-DH lesions appear to have dramatic effects on the adaptability of temporal associations, whereas VH lesions only affect the response levels. These data suggest that the DH is more closely related to the core timing mechanisms involved in duration encoding [34, 35, 36, 37, 38] and the VH is more closely related to motivation and context-dependent modulation of timing performance [39, 40]. In a series of experiments investigating the relationship between the DH lesions and peakinterval response, significant leftward (earlier) maximal responses were found when compared to sham-lesioned subjects [26]. Apart from the order of the surgery with respect to animals’ acquired instrumental responses, i.e. surgery first in [23, 24, 25, 26] and surgery last in [27], all studied found that DH lesions produced a leftward shift in peak time. The above brief summary of some experimental results on hippocampus lesions led us to formulating a novel theoretical hypothesis of a topological map of temporal memories stored in the hippocampus.

This paper advances the hypothesis that the long-term storage of temporal memories follows a topological map in the hippocampus, with longer durations stored towards dorsal hippocampus and shorter durations stored toward ventral hippocampus. The hypothesis was evaluated in the framework of two leading models of interval timing: SBF and SBF-ML. The predictions of this hypothesis match current experimental data on the effect of hippocampal lesions on time perception, and open new avenues for further experimental testing.

2. A New Hypothesis: A topological temporal map in the hippocampus

2.1. Assumptions: topological maps in the brain

To mimic theexperimentally-observed variability of the memorized criterion time T, we randomly generated a wide range of values around the desired criterion time T using a specified distribution. As an example, in Fig. 2 we generated a Gaussian (normal) distributed criteria around T = 10s (see [16, 17, 6, 21, 22]). As lesion studies suggested [23, 24, 25, 26, 27], we modeled the hippocampus (see Fig. 2A) storage as a spatially distributed map of the Gaussian representation of the reinforcement time values.

Figure 2: Modeling hippocampal lesions.

(A) A sketch of the three-dimensional organization of the rat’s hippocampus emphasizing the dorsal (DH) and ventral hippocampus (VH) areas; redrawn from [41]. The Gaussian distribution of criterion time is spatially mapped onto hippocampus leading to a topological representation of time with durations ordered from short to long uniformly distributed from ventral to dorsal areas. The center of the Gaussian distribution of memorized time is at the media line between the dorsal and ventral hippocampus. (B) During the training, multiple trials produce slightly different representations of the criterion time in the reference memory that overall lead to a continuous Gaussian distribution [6, 16, 17, 21, 22]. Hippocampal lesions removed memory cells in an unbalanced manner that biased the remaining memorized values towards lower values of criterion time (see dark shaded rectangle). (C) Due to the topological organization of the memory, lesions reduce the actual memory size and produce a non-symmetric memory of learned criterion time.

Based on the selected memory distribution (see Fig. 2B), different memory cells hold slightly different values of the criterion time in a topologically ordered map. For example, based on Fig. 2B, the relative frequency of memory allocation for T = 10 s is maximum possible, i.e. this criterion time will be stored in the largest possible number of memory cells.

As with any numerical implementation, the number of memory cells must be finite and, therefore, the continuous, smooth, distribution from Fig. 2B was replaced by a discrete counterpart (see Fig. 2C). This is why, according to Fig. 2C, the criterion time T = 10 s is stored in a large contiguous block of memory cells. Similarly, from Fig. 2B, a slightly shorter criterion time T = 9 s has a smaller fraction of assigned memory cells. As shown in Fig. 2C, the key assumption of our model of hippocampus memory organization is that the distribution of durations acquired during reinforcement trials when the criterion time is learned is (1) ordered, e.g. from low to high values, and (2) stored in successive memory locations to generate a topological map.

In our computational implementation of the topological map of hippocampus, memory lesions are represented by the light-shaded rectangle marked “lesion” that biases the originally symmetric criterion time representation storage centered on a criterion time of T = 10 s (see the continuous curve in Fig. 2B) towards values between (T_min, T_max) that include the criterion time. In Fig. 2B, a VH lesion left intact memory cells biased towards the longer durations end of the original, symmetric, Gauss distribution. We assumed that the abstract “center” of the distribution that corresponds to the memorized criterion time acquired during multiple reinforcement trials (see Fig. 2B and Fig. 2C) could be interpreted as the median line between the dorsal and ventral parts of the hippocampus (Fig. 2A). For example, lost memory cells due to DH lesions determine a non-symmetric distribution of memorized times biased towards shorter durations (see Fig. 2C) and, as a result, a smaller actual memory size than the one used durin the training. The post-lesion memory size (see the dark rectangles in Figs. 2B and C) width_post is related to the size of the hippocampus lesions quantified in experiments [23, 24, 25, 26, 27], i.e. % lesion size = 1 − width_post/width_pre. At the same time, the lesion “offset” with respect to the median line between the dorsal and ventral sides of the topological map (see Fig. 2B and Fig. 2C) determines the range of pre-lesion durations that will be used for post-lesion decision on interval timing and is related to what is called lesion location in experimental studies [23, 24, 25, 26, 27].

2.2. Corollary of topological map assumption – A discrete range of stored durations during hippocampal lesions

While highly efficient computational algorithms can generate (pseudo)random numbers according to a given distribution function, there are a few practical implementation details of hippocampal topological map that require careful evaluation in order to obtain an accurate prediction. For example, assuming that the criterion time T is generated according to a Gaussian (normal) distribution with mean T and standard deviation σ_T, we first found the range of the distribution. This is an important conceptual step towards a complete theoretical solution since the memory has a finite number of calls and, as a result, the lesions would only produce an output based on a finite subset of the randomly generated range. Although the post-lesion values of the criterion time are still part of a normal distribution, the distribution is skewed and, as a result, it would skew the output of the timing model. We must emphasize that the peak shift we predicted theoretically is purely due to the hypothesized topological organization of the hippocampus and has no relationship to clock model, accumulator model, or other conceptual pieces of SET model or its extensions.

In the subsequent mathematical derivation, we used the normal probability distribution function pdf_x of a random variable x with zero mean and unit standard deviation N(0,1) to mimic the reinforcement time distribution. Such a distribution could be transformed into a Gaussian pdf_z for the variable z with the mean T and the standard deviation σ_T by the following change of variables$x=(z-T)/{\sigma }_T$. Following [42], let us consider n samples x₁,x₂,… ,x_n from N(0,1). According to the extreme order statistics (see [42, 43, 44] and references therein), the probability distribution of the greatest values among n samples from N(0,1) is: $pd{f}_{x_{max}}=n\mathrm{\Phi }{\left(x_{max}\right)}^{n-1}\varphi \left(x_{max}\right)$, where $\varphi (x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$ is the pdf_x and $\mathrm{\Phi }(x)=\underset{-\infty }{\overset{x}{\int }}\frac{1}{\sqrt{2\pi }}{e}^{-u^2/2}du$ is the cumulative distribution function — (cdf) of N(0,1) [45].

Since the cdf of N(0,1) cannot be expressed using elementary functions, Φ(x) cannot be analytically integrated and the moments of the extreme order distribution are difficult to find analytically. One solution is based on a recursive evaluation of the cdf [46], which is limited to small samples [47]. Some studies used numerically generated tables of cdf for selected sample sizes [48]. Another, more general approach, is to provide explicit approximations for the greatest order distribution numerically in therms of Φ(x) (see [49], which has the problem of sample size-dependent accuracy, or [50], which has an estimation error as high as 8% for small samples). In the following, we used a somewhat better estimator proposed by Chen and Tyler [42], in which the expected value of the greatest order statistics for Gaussian samples is M = Φ^–1(0.5264^1/n), which has an accuracy of the order of 0.5%. In other words, M is the best estimate of the largest value out of the n values of the criterion time stored during the reinforced trials.

A wide range of analytic approximations of the cdf for Gaussian distribution were suggested. Among others, power series expansions with a narrow range (|x| < 4.2) and good accuracy (better than 10^–7) were suggested [51], or with an improved range [52]. In this study, we used a sigmoidal approximation (see Fig. 3A) for the cumulative distribution function that has the advantage of a wide range (|x| < 8) and good accuracy (better than 10^–5) [53]:

$$\mathrm{\Phi }(x)=\frac{1}{1+e^{-\sqrt{\pi }\left({\beta }_1{x}^5+{\beta }_2{x}^3+{\beta }_{3}x\right)}},$$ (1)

with β₁ = –0.0004406, β₂ = 0.0418198, and β₃ = 0.9000000.

Figure 3: Cumulative distribution function for Gaussian noise.

(A) Although the cdf of a Gauss distribution has analytical formula, a very good approximation is the sigmoidal (Eq. 1). If the hippocampus is topologically organized as in Fig. 2, a ventral lesion removes some memory cell that store durations below a minimum value T_min. Similarly, a dorsal lesion removes cells that store durations above a value T_max. The limits T_min and T_max have a complex nonlinear dependence on the lesion size, location, and the number of memory cells. However, T_max can be approximated by the linear expression $\frac{offset-widthpost/2}{N_{mem}}$ with less thsan 5% error (b). Similarly, the linear approximation$\frac{offset-widt{h}_{post}/2}{N_{mem}}$ of T_min gives an error that is less than 1%. Therefore, the nonlinear terms in the complex Eqs. 5 were neglected.

Based on the definition of the cdf and the hypothesis of a topological map as shown in Fig. 2B, in the case of a ventral lesion, the fraction n₁ of the memory cells holding values above T_min is:

$$1-n_1=\underset{T_{\min }}{\overset{M}{\int }}\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}dx=\mathrm{\Phi }(M)-1\mathrm{\Phi }\left(T_{\min }\right).$$ (2)

Similarly, in the case of a dorsal lesion shown in Fig. 2C, the fraction n₂ of the memory cells holding values below T_max is:

$$1-n_2=\underset{T_{max}}{\overset{M}{\int }}\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}dx=\mathrm{\Phi }(M)-\mathrm{\Phi }\left(T_{max}\right)$$ (3)

On the other hand, the fraction of cells removed from the pre-lesion memory with the size N_mem are related to the post-lesion memory window size width_post (lesion size) and the offset (lesion location) as follows:

$$n_1=\frac{N_{mem}/2+offset-widt{h}_{post}/2}{N_{mem}}$$ (4)

$$n_2=\frac{N_{mem}/2+offset+widt{h}_{post}/2}{N_{mem}}$$

By combining Eqs. (2), (3), and (4) we get the estimates of the post-lesion range of criterion time:

$$\mathrm{\Phi }\left(T_{min}\right)=\mathrm{\Phi }(M)-0.5+\frac{offset-widt{h}_{post}\text{/2}}{N_{mem}}={0.5264}^{1/N_{mem}}-0.5+\frac{offset-widt{h}_{post}/2}{N_{mem}},\mathrm{\Phi }\left(T_{max}\right)=\mathrm{\Phi }(M)-0.5+\frac{offset+widt{h}_{post}/2}{N_{mem}}={0.5264}^{1/N_{mem}}-0.5+\frac{offset+widt{h}_{post}/2}{N_{mem}}.$$ (5)

The above estimations for T_min for the ventral lesions (Fig. 3B) and T_max the dorsal lesions (Fig. 3C) are determined by the pre-lesion memory size N_mem, the post-lesion memory size width_post, which is related to lesion size % lesion size = 1 — width_post/N_mem, and the offset (location) of the memory lesion with respect to the “center” (or the median location between dorsal and ventral hippocampus) of the pre-lesion criterion time distribution (see Fig. 2). Using the estimates from (Eqs. 5) and the sigmoidal approximation of cdf given by (Eq. 1), we numerically estimate the post-lesion range (T_min,T_max). For N_mem = 400, width_post ≤ 200, and offset ≤ 100 (see Fig. 3), we found that the range (T_min, T_max) is quite well approximated by the linear part (last term) in (Eqs. 5), i.e.

$$T_{min}\approx a\frac{offset-widt{h}_{post}/2}{N_{mem}},T_{max}\approx b\frac{offset+widt{h}_{post}/2}{N_{mem}},$$ (6)

with a = 2.93083 and b = 2.57356. The absolute error between the actual T_max from (Eqs. 5) and the approximate linear expression from (Eqs. 7) is bellow 5% (see Fig. 3B). Similarly, the absolute error for T_min shown in Fig. 3B is bellow 1%. As a result, for all subsequent calculations we only used the linear approximations of T_min and T_max given by (Eqs. 7).

2.3. Corollary of topological map assumption - A predicted peak shift in the mean memorized duration with hippocampal lesions

The above realistic estimates of the post-lesion range of stored criteria (T_min, T_max), allowed the calculation of good estimates for the peak shift of the output due to memory lesions. Indeed, for a Gaussian pdf,$\varphi (x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$ , the most likely estimate for the center of the distribution is

$$\overline{x}=\underset{T_{min}}{\overset{T_{\max }}{\int }}x\varphi (x)dx=\frac{T_{max}^2-T_{min}^2}{4\sqrt{2\pi }}.$$ (7)

Using the linear approximation of the range given by Eq. (7) and assuming that the numerically-estimated constants a and b are identical, the above estimate for the center of the post-lesion distribution becomes:

$$\overline{x}\propto \frac{offset}{N_{mem}}\times \frac{widt{h}_{post}}{N_{mem}}.$$ (8)

As we previously showed (see [54] for a review), the new peak of the output is shifted to$\overline{T}=T\left(1+{\gamma }_T\right)$ , where${\gamma }_T\propto \overline{x}$ As we notice form (Eq. 9), if the postlesion window has no off set with respect to the pre-lesion (see Fig. 2), then there is no shift of the peak response compared to pre-lesion. In other words, it is possible to perform hippocampus lesions that do not change the peak location of the output. Although the result makes sense intuitively, we must emphasize that in reality even with no offset the post-lesion memory length width_post may produce some peak shift since (Eq. 9) is only approximately valid. Among other assumptions, identical coefficients in the linearized approximations (a = b) in (Eq. 7) is not truly fulfilled and, therefore produces quadratic terms in (Eq. 8) that would slightly shift the peak even without any offset.

As a result of Eq. (9), we predicted mathematically that the peak shift due to lesions is proportional to the product of the normalized offset (off set/N_mem), i.e. lesion location with respect to median line between ventral and dorsal regions of the hippocampus, and the normalized post-lesion memory size (width_post/N_mem), which is related to the size of hippocampal lesion.

It must be stressed out that the above theoretical results considered a Gaussian with zero mean and unit standard deviation N(0,1), which could be rescaled to a Gaussian for any criterion time T and standard deviation according to the transformation x = (z — T)/σ_T. This is important because a wider Gaussian distribution (larger standard deviation) of memorized times also impacts the peak shift. For example, for a criterion time of T = 10s and a standard deviation of σ_T = 0.1T, a 34% lesion to the left of the Gaussian peak coved durations from 9 s to 10 s in our model of the topological map of the hippocampus. How-ever, if the standard deviation is double, then the same percentage lesion of the topological map covered durations from 8 s to 10 s, which obviously would shift the peak of the output to lower values compared to the previous case. This observation also results from the mathematical transformation of random variables between the x variable of N(0,1) and the new variable z for N(T, σ_T), i.e. x = (z — T)/σ_T. Indeed, the new random variable for arbitrary criterion times T and standard deviations σ_T is z = T + xσ_T, which apart from a temporal shift by a constant T shows that the values of the random variable x with N(0, 1) must be multiplied by the standard deviation σ_T to get the correct effect on the scaled random variable z that obeys a Gaussian distribution with mean T and standard deviation σ_T. In other words, the peak shift from Eq. 9 for N(0,1) generalizes to:

$$\overline{z}\propto {\sigma }_T\frac{offset}{N_{mem}}\times \frac{widt{h}_{post}}{N_{mem}},$$ (9)

for a Gaussian topological map N(T,σ_T). This is the key theoretical result of this study and shows the explicit mathematical formula for K* that could be tested experimentally.

We carried out numerical simulations and found that the above theoretical predictions are accurate. We also found very good agreement between our numerical predictions regarding the effect of lesion size on the peak shift and the published hippocampus lesion results.

3. Numerical Verification of Theoretical Predictions based on a Topological Map of Hippocampus

The above theoretical prediction of the memory lesions effect on the shift of the peak-interval timing shown in Eq. 10 provides the first quantitative estimation of the “memory translation constant” K* by combining (1) the assumed Gaussian representation of the reinforcement time in the reference memory (see [16, 17, 6, 21, 22]) with (2) the experimental observations of precise and reproducible peak-interval shifts correlated with the spatial location of hippocampus lesions (see [23, 24, 25, 26, 27]) into a novel hypothesis of a hippocampal topological map of durations. While the above theoretical predictions based on a topological map assumption are independent of the particular implementation of the timing model, to check our predictions we used our previous implementation of the SBF-ML (SBF model with Morris-Lecar model neurons) [15, 54, 55] for three different criteria T = 10 s, 20 s and, respectively, 30 s and a constant memory variance of 10%. We used four different pre-lesion memory sizes of N_mem = 200, 300, 400, and, respectively, 500 memory cells. For the post-lesion, we used three values of width_post = 100, 200, and, respectively, 300 memory cells. We numerically investigated the effect of the off set, i.e. lesion location, (see Fig. 2) of the contiguous window of post-lesion memory cells with respect to the criterion time both on the peak of the output (see Fig. 4) and the coefficient of variation (see Fig. 5). All peak shifts were normalized with respect to the corresponding criterion time (see Fig. 4).

Figure 4: Peak shift versus lesion location.

Relative peak shift (in % of the corresponding criterion time) due to post-lesion memory window offset, i.e lesion location relative to the median line between dorsal and ventral topological map of hippocampus, for different prelesion (N_mem) and post-lesion (width_post) memory sizes. The zero “offset” corresponds to the medial line between the dorsal and the ventral sides of hippocampal topological map (see Fig. 2).

Figure 5: Coefficient of variation versus normalized memory offset.

The CV due to post-lesion memory window offset (lesion location) for different pre-lesion (N_mem) and post-lesion (width_post) memory sizes shows a somewhat nonlinear dependence on post-lesion memory offset.

We noticed from our numerical simulations that shifting the post-lesion window to the left of the criterion time, i.e. a dorsal lesion according to Fig. 2 topological map, led to a leftward proportional shift of the peak output and vice-versa (see Fig. 4). These numerical observations are in agreement with our theoretical prediction based on (Eq. 9) which suggested that the peak shift should be proportional to the normalized post-lesion memory offset (off set/N_mem).

The slopes for different combinations of pre-lesion memory size N_mem and post-lesion memory size width_post for three different values of the criterion time T = 10 s, 20 s, and 30 s are given in Table 1. The slopes correspond to the panels shown in Fig. 4. Using the data from Table 1, we also noticed that the slopes are proportional to the normalized post-lesion memory size (width_post/N_mem). Therefore, combining the numerical observations from Fig. 4, i.e. the peak shift is proportional to off set/N_mem, and the fact that the corresponding slopes of the peak shift are proportional to width_post/N_mem, it results that the peak shift is in fact proportional to the product of the normalized offsets (lesion location) and the normalized post-lesion memory sizes (lesion size) as predicted theoretically by (Eq. 9).

Table 1:

Slopes of the normalized peak shift (time shift/criterion time) versus normalized offset (off set/N_mem) for different pre-lesion memory sizes (N_mem) and post-lesion memory width_post with three different criteria T = 10 s, 20 s, and 30 s.

Nmem	widthpost	T = 10s	T =20 s	T =30 s
200	100	29.42	28.76	28.16
300	100	30.63	32.13	30.75
300	200	33.3	29.91	34.86
400	200	28.56	31.4	28.96
400	300	34.48	37.16	35.2
500	300	29.15	25.55	24.15

4. Comparison of Theoretical and Numerical Results based on a Topological Map of Hippocampus and Some Experimental Data on Hippocampus Lesions

Although we did not carry out our own behavioral experiments to check the theoretical (see Eq. 10) and numerical predictions based on hippocampus topological map hypothesis, we did search the literature for published experimental data on hippocampus lesion effects. Although our search is far from being exhaustive, we found that in Tam et al. [23] the authors used a criterion time of T =15 s and they measured a peak shift to about 10 s (see Fig. 6 in [23]), which corresponds to a 33% shift, for a mean damage of 38%. Although the authors performed DH lesions at slightly different locations, they took an average of lesion size and, therefore, the information about lesion location was lost. In our numerical results (see Fig.4), the closest lesion sizes to their experiments are reported in panels A2 (30% lesion size) and, respectively, A1 and B2 (50% lesion size). In all cases, the maximum possible peak shift with a standard deviation σ_T = 0.1T was 10%. To reach a 33% peak shift as observed in the experiments done by Tam et al. [23], according to our theoretical prediction from Eq. 10, the standard deviation of the memorized criteria should be at least three times larger than in our simulations, i.e. for a criterion time of T =15 s they probably could have observed a standard deviation of memorized times of σ_T ≥ 0.3T =4.5 s. In a recent peak-interval timing study on rats [56] the spread of the response was defined as the difference between the time it first reached half the maximal rate and the last time at which it descended to the half maximal rate, which for a Gaussian distribution is called the full width at half maximum (FWHM). This value of the FWHM is related to standard deviation as follows: FWHM ≈ 2.3548a. Therefore, based on the results from Table 1 in [56] we found that for a midpoint (peak) at 32.82 s the FWHM = 23.65 s, which corresponds to a standard deviation of 10 s. Similarly for T =30.09 s they found FWHM = 29.24 s, which gives σ_T =12.4 s, etc. These experimental data suggest that σ_T = (0.3 — 0.4)T in rats [56]. Similar results were published in Tam et al. [23], which based on their Table 4 show a wide range from the peak time at 45.41 s and FWHM = 27.73 s to a peak time of 27.5 s and FWHM =27.67 s. As a result, their standard deviation is in the range of σ_T = (0.26 — 0.43)T. Therefore, experimental results seem to indicate that the standard deviation in rats is larger than the 10% of the criterion time used in our simulations and can cover a broad range from 26% to 42% of the criterion time as in [23]. This means that the peak shifts found in our numerical simulations (see Fig. 4) must be multiplied by a factor between 2.6 and 4.2. It results that in our of topological map model we should expect maximal peak shifts between 26 s and 42 s, which is consistent with the experimentally observed 33% peak shift reported in [23] for a mean 38% lesion size of DH.

In a different study with a criterion time of 40 s on rats [24] it was found that DH lesions shift the peak responses over a broad range from a peak at 36.9 s and a spread of 23.6 s, i.e. σ_T = 0.27T to a peak at 36.7 s and a spread of 50.4 s, i.e. σ_T = 0.58T. As a result, our numerical findings regarding the peak shift from Fig. 4 must be scaled by a factor between 2.7 and 5.8. In these experiments they reported that hippocampal damage was 38% of total hippocampal volume (range: 15–45%) [24], which would match the lesion sizes we show in panels A2 (30% lesion size) and, respectively, A1 and B2 (50% lesion size) of Fig. 4.

We also investigated the change in the standard deviation of the output with the post-lesion memory size and the lesion offset (see Fig. 5). Scalar property asserts that the coefficient of variation (CV), i.e. the ratio of the standard deviation of the output by the peak criterion time, should be constant [3]. We found that the CV is not quite constant, although in most cases the spread of the numerical data was quite large to allow a definite answer.

Using experimental data from the same studies, for example [24], we computed the CV from the DH lesion spread shown in their Fig. 6C and the peak times shown in Fig. 6A. Using the same two examples as above, the coefficient of variation was between 0.27 and 0.58. In our numerical simulations, we notice that for a lesion size of 33% (see panel A2 in Fig. 5) the CV smoothly varies from 0.05 to 0.1 depending on lesion location. Also, for 50% lesion sizes (see panels A1 and B2 in Fig. 5) we found that the coefficient of variation smoothly varies form 0.07 to 0.12. Our numerical simulations were performed with a standard deviation of the Gaussian topological map of 10% of the criterion time. The above brief summary of some experimentally published data on hippocampus lesion suggest an actual standard deviation of the output between 27% and 58%. After appropriate scaling of our numerical results shown in Fig. 4 to accomodate a larger σ_T variance according to Eq. 10, our numerical simulations provided results consistent with published experiments on hippocampus lesions.

5. Discussion

We predicted theoretically that the peak shift due to hippocampal lesions in this newly proposed hypothesis of a topological organization of hippocampus is proportional with both the normalized post-lesion memory size an the spatial location (offset) of the lesion. We found numerically that the relative peak shift is proportional to both the post-lesion memory size and its offset, although there are a few caveats to consider (see Fig. 4). For example, we found theoretically that the first approximation of the peak shift is a quadratic form that indeed involves the product of the post-lesion memory size and its offset, but it also contains the squares of the post-lesion memory size, respectively, the square of the offset. As a result, we also predict that even in the absence of a spatial offset of the lesion there could be a peak shift that is solely determined by the normalized post-lesion memory size. The fact that we did not observe it in our numerical simulations could be due to its small magnitude compared to the product of the post-lesion memory size and its offset, which dominated the temporal shift.

We also computed the coefficient of variation (CV) to check if scalar property of interval timing still holds, i.e. check if CV is constant. We found that the CV is changing with both the post-lesion memory size and its offset (see Fig. 5). However, the caveat here is that the standard deviation of the estimates is quite large and a significantly larger set of numerical simulations would be necessary in order to definitely settle this issue. However, based on the available data we predict that the scalar property does not hold in the case of hippocampal lesions.

We also found that our theoretical predictions of the peak-interval shift match previously published hippocampus lesions experimental data. We also predicted that the lesion location is important in regard to peak shift, i.e. a more dorsal lesion should produce a more pronounced peak shift compared to a lesion of the DH that is closer to the median line between the DH and VH. Since the experimental results we found in the literature averaged the effect of peak shift over different locations, we could not compare our theoretical predictions agains experiments. However, the comparative study we carried out between our predictions and hippocampus lesion experiments suggests that probably the experimental data are already available and only need to be reported as peak shifts versus lesion location instead of averaging over all lesion locations.

Similarities with other computational models of interval timing.

During the over four decades since the seminal work of Gibbon and Church [5, 6, 16, 17, 57] on the SET, many new models were suggested. Some studies separated the conceptual elements of SET into first-order principles, such as changes in clock speed and temporal memories storage, and second-order principles, such as timescale invariance [20] and analyzed them separately. Other studies focused on specific neuroanatomical structures and investigated their role in interval timing, such as cortico-striatal-thalamo-cortical motor circuit [58]. Some theoretical models departed from SET and used multiple oscillator that produce beats or synchronous patterns among the population of oscillators [59, 60]. Theoretical and numerical studies may assign the timing abilities to the entire neural network [61], attribute it neuronal elements with a broad spectrum of time constants (labeled line models) [62], or encode durations in a population of neurons active at any specific point in time [63, 4]. A recent extension of the nonlinear rate models was introduced by Shea-Brown et al. [64] and allowed the extension of SET accumulator to account for migration effect and uniform overestimation trends in Parkinson’s disease interval timing data [65, 66]. In particular, the curvilinear accumulation model of Shea-Brown et al. [64] uses an idealized recurrent neural network with two adjustable parameters: the neural feedback and its the external drive to the population. The firing rate of this integrable model determined the criterion time as the first duration at which the firing rate of the network reaches a preset threshold. From a neurobiological perspective, the model is closely related to the striatum comparator function that decides on a given output based on a learned threshold [5, 6, 16, 17, 57]. There is in vivo electrophysiology support for the population models that show that some neurons exhibit linear change in firing rate as time elapses (see for example [67, 68, 69, 70]). Although graded firing rates usually peak at the time of an anticipated response, monotonic firing rate were also reported [71]. The mathematical model of a topological map of hippocampus that we introduced here mimics the hippocampal memory of temporal durations and the effect of hippocampal lesions. Our model does not use an accumulator and it actually uses a numerical implementation that belongs to the class of the beats among population of oscillators [59, 60].

Potential neuromporphic implementation of memory maps.

Given the continuous expansion of neuromorphic applications, it may be possible to even implement this novel topological map hypothesis in the design of a memory chip. For an efficient neuromporhic implementation, it would be useful to better understand and use some of the optimization principles that have been discovered for large-scale integrated memory circuits. For example, it has been found experimentally that the retention time of a dynamic random-access memory (DRAM) cell is strongly affected by the value stored both in that cell and in nearby cells due to circuit-level crosstalk effects [72, 73, 74]. Therefore, storing similar values in nearby cells prevents the occurance of large gradients between cells and improve retention time. In addition, electrical coupling between adjacent bitlines creates noise on each bitline, noise that depends on the voltages of nearby bitlines. Since the goal is to store and retrieve accurate values for criterion time from each memory cell, the noise experienced by each bitline that is determined by the values stored in nearby cells must be minimized [74, 75]. Such interferences could lead to hard faults whose manifestation depends on the data stored in nearby memory cells, called neighborhood pattern-sensitive memory faults [76]. Another important reason for a topological organization of memory chips is to reduce the standby power of such circuits [77].

Among other advantages, a topological map model minimizes “memory leakage” and allows efficient coding of information. The “memory leakage” produces a smearing of values stored in the memory and is often used for the purpose of simulating memory degradation. Since usually memory degradation is modeled as a diffusion process, having adjacent memory cells with the smallest possible gradient (difference between criterion time stored values in adjacent cells) makes the diffusion and memory smearing a slow an homogeneous (similar smearing across different cells) process. The spatially ordered distribution of criterion times is also advantageous for other methods of information coding. For example, here we assumed that the information is coded as numbers (criterion times) stored in memory cells. Other models could use the rate of change of a reference criterion time, e.g. by how much T = 10 s changed. Such a model would have the advantage of using a minimum number of bits to store only what and when the state of the system changes rather than storing all possible states.

6. Conclusions

Besides derived an explicit, experimentally testable formula for the “memory translation constant” K^*, this study also shows that K^* is expressed in terms of relative size of lesion and its relative position to the median line between DH and VH. Based on a series of previous publications, showing that the standard deviation of the output scales with the standard deviation of the noise [2, 13, 54, 55, 78], we further augment the peak shift formula by including in our predictions the contribution of random noise on stored memory values (see also Eq. 10). We predicted that regardless the number of memory cells, N_mem, in the hippocampus each organism should be affected in the same manner by hippocampal lesions. This is because the temporal shift in our topological map model only depends on the relative size and the relative location of the lesion with respect to the ventral-dorsal median line. These are important and experimentally testable predictions that need further experimental investigation.