Modeling Time Cell Neuron-Level Dynamics

Abstract

The timing of intervals in both humans and nonhuman animals is susceptible to imprecision, leading to trial-to-trial variations in timed behaviors. A notable statistical characteristic of this timing behavior is the linear increase in the standard deviation of timed responses as the intervals become longer, known as time-scale invariance. Although the exact mechanism responsible for this property is not yet fully understood, “time cells” have emerged as a potential mechanism, particularly in relation to interval timing within the episodic memory system. These specialized networks of neurons exhibit sequential firing, with each neuron becoming active during a specific phase of the timed experiment. Notably, the resulting “ramping wave” decelerates during later phases of the timed event, and the duration of cell activity increases over time. In this paper, we propose a new model of a biological neural network composed of time cells, utilizing integrate-and-fire neurons with slow after-hyperpolarization currents and varying resting membrane potentials. This proposed network does not rely on any specific network architecture apart from self-excitation. Remarkably, the network successfully reproduces the experimentally observed ramping activity, with the standard deviation of spike times among the participating neurons increasing linearly with the average spike time. Furthermore, the results of the model are consistent with the experimental findings, providing additional evidence for its accuracy and validity.

Introduction

Animals, including humans, have the remarkable ability to learn, remember, and organize their responses based on time intervals ranging from seconds to minutes. This fundamental capability, known as interval timing, is widespread among species and has been conserved throughout evolution. Integrating interval timing into behaviors related to foraging and defense is crucial for an animal’s survival in natural environments. It allows them to adapt their responses by synchronizing them with the statistical properties of their surroundings (Buhusi & Meck , 2005; Merchant & Lafuente , 2014). Temporal information also plays a vital role in the formation of episodic or episodic-like memories, and it can be acquired even after a single exposure or experience (Bevins & Ayres , 1995).

The psychophysical properties of interval timing provide insights into the underlying mechanisms and the neural networks involved in temporal processing within the brain. One notable psychophysical property of interval timing is time-scale invariance, also known as the scalar property. This property is characterized by the coefficient of variation (CV) of response times converging to a constant value as the time intervals increase. The coefficient of variation is calculated by dividing the standard deviation of response times by the mean response time (e.g., see Gibbon et al., 1984; Oprisan & Buhusi, 2014; Simen et al., 2011; Zeki & Balcı, 2019).

The modeling of the scalar property presents a significant challenge as it cannot be accurately captured by simply summing individual random variables. This is because the variability of the sum does not increase proportionally with the number of variables added. As a result, the most straightforward approach to modeling subjective time fails to account for the observed time-scale invariance. Although numerous models have been developed to address the scalar property, the availability of neurobiological models that can effectively capture the properties of timing behaviors is limited. Furthermore, these neurobiological models often rely on specific architectures for the connections among the neurons within the network (e.g., see Addyman et al., 2016; Balcı & Simen, 2016; Matell & Meck, 2000; Meck, 2003; Simen et al., 2011; Zeki & Balcı, 2020).

Recent research has provided evidence for the existence of “time cells” (TC) in various brain regions, including the hippocampus, entorhinal cortex, striatum, and cerebellum. These time cells function by sequentially encoding and tracking the passage of time, similar to how place cells in the hippocampus encode spatial information (e.g., see Eichenbaum, 2014; Kraus et al., 2013; MacDonald et al., 2011; Pastalkova et al., 2008; Salz et al., 2016; Tiganj et al., 2017; Umbach et al., 2020). While the ensembles of time cells have been well characterized in certain memory-related experiments (e.g., see MacDonald et al., 2011; Pastalkova et al., 2008), it remains unclear whether the same set of cells are involved in other experiments involving timing. In a study by Lusk et al. (2016), the authors explored the integration of different timing systems, including the contribution of time cells from various brain regions.

In vivo neurophysiological studies have provided evidence that time cells exhibit specific firing patterns at distinct epochs of a timed event and demonstrate a topographical representation of time (e.g., see Eichenbaum, 2014; Kraus et al., 2013; Salz et al., 2016; Tiganj et al., 2017). The activity of these cells is observed through their averaged trial activity, which is characterized by a Gaussian-shaped time field. Importantly, the width of these firing fields increases as time elapses. When the time cells are organized based on their peak firing times, a pattern emerges that can be mapped onto a corresponding behavioral response. The fact that the width of the time fields increases with the peak firing time indicates a decrease in precision for longer time intervals.

In this study, we introduce a biophysical neural network model of time cells that avoids complex architectural assumptions, such as chain-like connections between neurons, and synaptic connection assumptions beyond self-excitation (positive feedback from each spiking activity). The model accurately represents the patterned activity of time cells by having neurons fire at specific epochs of the timed activity, determined by their resting potential levels. The ordering of cells based on their resting potential levels aligns with experimental observations, where cells with lower resting membrane potentials fire later and exhibit wider firing field widths, similar to observed time cells.

Crucially, our model demonstrates the emergence of the scalar property, in line with the psychophysics of interval timing, as the variability scales linearly with time. Additionally, our model neurons possess the ability to adjust their peak activity window based on the initial values of the slow after-hyperpolarization current, reflecting the flexibility known as “temporal scaling” observed in experimental studies.

To validate our model, we conducted experiments similar to those performed by Tiganj et al. (2017). Furthermore, we compared the model data with experimental data from Swanton and Matell (2011) and Crystal et al. (2003). The results of both comparisons consistently show a strong agreement between the model outcomes and the experimental outcomes, providing further support for the accuracy and validity of our model.

Materials and Methods

Scalar Invariance and Time Cells

As mentioned previously, a significant characteristic of timing behaviors is the scalar property, which refers to the linear increase in imprecision with longer time intervals (e.g., see Balcı & Simen, 2016; Buhusi & Meck, 2005; Gibbon et al., 1984; Meck, 2003; Simen et al., 2013). In other words, when target time intervals change, the ratio of the mean of timed responses or temporal judgments to the standard deviation of those responses or judgments remains relatively constant. This phenomenon is observed in both human experiments (when chronometric behaviors are suppressed) and experiments involving nonhuman animals.

The time-scale invariance property is challenging to explain in an additive manner, as the standard deviation of the sum of random variables scales with the square root of the individual standard deviations. Specifically, if we add N random variables with a mean of M and a standard deviation of S, the mean of the sum will be $N·M$, but the standard deviation of the sum will be $\sqrt{N}·S$. This simple observation necessitates the need for additional assumptions or the development of new modeling approaches to account for the psychophysics of interval timing.

Recent experimental studies have provided evidence for the existence of time cells in the hippocampus and entorhinal cortex, which play a role in tracking time as part of episodic-like memory (e.g., see Crystal, 2021; Hampton & Schwartz, 2004). These cells exhibit specific firing patterns during different epochs of temporally structured experiments (e.g., see Eichenbaum, 2014; Kraus et al., 2013; Salz et al., 2016; Tiganj et al., 2017; Umbach et al., 2020). Notably, the speed at which the neuronal activity changes over time diminishes, and the cells exhibit prolonged activity during the later stages of the delay period (see also Mello et al., 2015 and Tiganj et al., 2017 for similar patterns in striatum and mPFC, respectively).

The precise biophysical mechanisms underlying the observed behavior of time cells are still unknown. It is possible that these patterns of activity arise from intrinsic properties of the network without requiring significant neural plasticity or extensive training. The presence of ramping activity even in the absence of extensive training suggests that these patterns may be a result of the intrinsic dynamics of the network rather than being solely dependent on functional neural plasticity.

In this study, we deviate from conventional approaches, including those employed in our previous models, that rely on specific network architectures or decay functions. By focusing on the interplay between resting membrane potentials and after-hyperpolarization currents, we aim to capture the essential dynamics of time cell activity while avoiding the need for specific architectural assumptions.

Methods: Network and Model

Time Cells

In our study, we focus on a network consisting of 50 integrate-and-fire excitatory neurons. The network is characterized by self-excitatory synaptic connections, and the dynamics of the membrane potential are governed by the equation:

$$\begin{array}{c}\hfill C_m\frac{\mathit{dV}}{\mathit{dt}}=-(I_L+I_{\mathit{sAHP}})+I_{\mathit{syn}}+I_{\mathit{app}}.\end{array}$$

Here, $C_m$ represents the membrane capacitance, which has a value of $1\mu F/c{m}^2$. The leak current, denoted by $I_L$, is given by the equation $I_L=g_L(V-V_L)$, where $g_L$ represents the maximal conductance of the leak current and has a value of $1\mu S$, and $V_L$ represents the leak reversal potential.

Neurons are assumed to generate an action potential when the membrane potential reaches a threshold level denoted as $V_{\text{thresh}}=-50,\text{mV}$. Once a spike is generated, the membrane potential gets back to reset level represented by $V_{\text{reset}}=-70,\text{mV}$.

As a crucial feature of this model, we assume that the resting potential values of the network neurons are not identical. To achieve the differential resting membrane potential values, we assumed that the leak reversal membrane potential values are uniformly distributed between $-65$ and $-73mV$.

The equation for the intrinsic slow after-hyperpolarization ($I_{\mathit{sAHP}}$) current is given by

$$\begin{array}{c}\hfill I_{\mathit{sAHP}}=g_{\mathit{sAHP}}m(V-V_K),\end{array}$$

where $g_{\mathit{sAHP}}$ is the conductance parameter for the slow after-hyperpolarization current, V represents the membrane potential, and $V_K$ is the reversal potential for the potassium ion ($K^{+}$). The sAHP current parameters are specified as $g_{\mathit{sAHP}}=4\mu S$ and $V_K=-90V$.

The activation variable m represents the gating variable for the sAHP current. It increases by a value of 0.3 with each spike and decays exponentially over time with a rate of change given by $dm/dt=-{\beta }_m·m$. The parameter ${\beta }_m$ determines the rate of decay for m and is set to ${\beta }_m=2.5\times {10}^{-5},{\text{ms}}^{-1}$.

The sAHP (slow after-hyperpolarization) current is recognized for its activation following the firing of multiple action potentials within a neuron (e.g., see Bond et al., 1999; Kim and Mccormick, 1998; Sah & Bekkers, 1996; Tombaugh et al., 2005). This current is activated by the influx of calcium into the cell, which subsequently triggers calcium-dependent, voltage-independent hyperpolarizing potassium currents (makes membrane potential more negative and thereby reduces the likelihood of action potential generation). One notable characteristic of the sAHP current is its extended decay time period: it activates within a few tens of milliseconds and exhibits an exponential decay with a time course lasting seconds (e.g., see Gerlach et al., 2004; Kim et al., 2012; Kim and Mccormick, 1998; Larsson, 2013; Tombaugh et al., 2005). By inducing a prolonged hyperpolarization, the sAHP current can effectively regulate neural excitability and prevent excessive firing of the neuron.

The presence of the sAHP current has been demonstrated in various brain regions, including the medial prefrontal cortex (mPFC) (e.g., see Fontanez-Nuin et al., 2011; Satake et al., 2008), hippocampus (e.g., see Sah & Bekkers, 1996; Storm, 1987; Tzingounis et al., 2007), and striatum (e.g., see Schulz & Reynolds, 2013; Wilson & Goldberg, 2006), where time cells have been reported.

Synaptic Currents

The current $I_{\mathit{syn}}$ that represents self-excitation of the network neurons is given by

$$\begin{array}{c}\hfill I_{\mathit{syn}}=g_{\mathit{exc}}s(V-E_{\mathit{AMPA}})\end{array}$$

with the maximal conductance $g_{\mathit{exc}}=250\mu S$ and the reversal potential, $E_{\mathit{AMPA}}=0mV$. Synaptic variable s is set to 1 after each spike and decays according to the equation

$$s^{\prime }=-{\beta }_{e}s$$

where ${\beta }_e=0.1m{s}^{-1}$.

In our model, we utilize self-excitation, which is a type of autapse, as a mechanism to strengthen the firing of the time cells (e.g., see Herrmann & Klaus, 2004; Ikeda & Bekkers, 2006; Ma et al., 2015; Wang et al., 2017). While autapses do exist in the neural system, they are not as prevalent in the brain to fully justify their usage in our model from a biological perspective. However, in our work, the interpretation is that each individual neuron represents a cluster of neurons with strong feedback excitatory connections. This concept aligns with the observation that excitatory neurons with dense connections tend to exhibit synchronized firing (e.g., see Buhl et al., 1998; Hahnloser, 1998; Tegnér et al., 2002; Wang, 1999). Therefore, the activity of these clustered neurons can effectively reinforce itself through the self-excitation mechanism employed in our model.

Network Geometry

We simulated a network of 50 TCs with self-excitation. In our model, we did not make any assumptions about the presence of synaptic connections among the TC neurons.

Noise

Network simulations were conducted by assigning the leak current reversal potential values as normally distributed random variables. This was achieved by adding random variables with a mean of 0 and a standard deviation of 0.45 to the resting potential values ($V_L$) of each time cell in the network. The inherent consequence of this variability in resting membrane potentials is the resulting variation in spike threshold levels among the time cells (e.g., see Fontaine et al., 2014; McKhann et al., 1997; Moutard et al., 2015).

Bisection Task Network

In order to simulate the bisection task experiment, two response units (RU) are integrated into the TC network, representing the short and long responses, respectively. The membrane potential equation for the RU neurons is defined as follows:

$$\begin{array}{c}\hfill C_m\frac{\mathit{dV}}{\mathit{dt}}=-I_L+I_{\mathit{syn}}.\end{array}$$

Here, the leak current $I_L$ shares the same characteristics and parameters as the TCs. The membrane capacitance $C_m$ is set to $1\mu F/c{m}^2$.

Synaptic Currents

The synaptic current $I_{\mathit{syn}}$ comprises two main components:

1. Inhibitory current ($I_{\mathit{inh}}$): This current represents lateral inhibition between the RU neurons for decision-making purposes and is described as

   $$\begin{array}{c}\hfill I_{\mathit{inh}}=g_{\mathit{inh}}{s}_i(V-E_{\mathit{GABA}}).\end{array}$$

   Here, $g_{\mathit{inh}}$ is the maximal conductance and is set to $2\mu S$, while $E_{\mathit{GABA}}$ is the reversal potential at $-90mV$. The synaptic variable $s_i$ is set to 1 after each spike and decays according to the equation:

   $$\begin{array}{c}\hfill s_i^{\prime }=-{\beta }_i{s}_i\end{array}$$

   where the decay rate ${\beta }_i$ is $0.1m{s}^{-1}$.
2. Excitatory current ($I_{\mathit{exc}}$): This current represents the sum of all excitatory synaptic currents originating from TCs and targeting RU neurons. It is defined as

   $$\begin{array}{c}\hfill I_{\mathit{exc}}=\sum g_{\mathit{rexc}}s(V-E_{\mathit{AMPA}}).\end{array}$$

   Here, s represents the synaptic variable, and $E_{\mathit{AMPA}}$ is the maximal reversal potential, both of which have been previously defined for TCs. The learning process for short and long reference durations involves strengthening the excitatory synaptic connections between TCs and RUs. This reinforcement of synapses occurs by incrementing the excitatory conductances after each TC spike that falls within a 500 ms window following either the short or long reference durations. The excitatory conductance $g_{\mathit{rexc}}$ is calculated as follows:

   $$\begin{array}{c}\hfill g_{\mathit{rexc}}=g_0+ndg.\end{array}$$

   Here, the initial excitatory conductance $g_0$ is set to $0.1\mu S$, and the incremental value dg is $0.003\mu S$, and n represents the number of reinforced trials for that particular TC. This process is repeated for a total of 300 learning trials.

Results

The Interplay Between the Slow After-Hyperpolarization Current (sAHP) and Self-Excitation

Figure 1 illustrates the impact of the sAHP current on the firing behavior of a single time cell. In the presence of applied direct current, the cell initially generates action potentials. However, over time, the frequency of the spikes gradually decreases. This adaptation in the firing pattern is primarily due to the activation of the sAHP current. The sAHP current acts as a hyperpolarizing current, causing the membrane potential to become more negative as its activation increases.

Fig. 1.

Activation of sAHP current with spike activity: In response to input, sAHP current is triggered with each spike and leads to a reduction in the firing frequency of the neuron. The sAHP current exerts its effect by inducing a hyperpolarization of the membrane potential, causing it to become more negative

Figure 2 presents the results of simulations conducted with and without self-excitation. It is important to note that in these simulations, the sAHP activation level is set to 1.

Fig. 2.

The role of self-excitation current: A time cell with pre-activated sAHP current is simulated with square pulse input both with and without self-excitation. The cell experiences a delay in spiking due to the decay of the sAHP-induced hyperpolarization. Once the membrane potential reaches the firing threshold, a cell with self-excitation is capable of producing multiple spikes through positive feedback from the excitation, which in turn activates the sAHP current. Following a burst of spikes, the cell eventually becomes silent. In contrast, a cell without self-excitation is unable to sustain high-frequency spiking

In the presence of input stimulation, the cell is unable to fire immediately due to the hyperpolarizing effect of the sAHP current. Eventually, as the sAHP current decays, the cell becomes capable of initiating firing. In the presence of self-excitation, which depolarizes the membrane potential (making it more positive), the cell is able to sustain firing through a positive feedback loop. However, as the firing persists, the sAHP current becomes further activated, resulting in the termination of firing. Once the firing ceases, the cell enters a refractory period where it is unable to fire for an extended duration due to the high activation of the sAHP current.

In contrast, in the absence of self-excitation, once the cell initiates spiking, it cannot maintain firing due to the activation of the sAHP current following each spike. The cell must wait for the sAHP current to deactivate before it can fire again. Consequently, the behavior of the cell following the initial spike can be characterized as low-frequency intermittent spiking.

Figure 3 demonstrates the impact of differences in resting membrane potentials in the presence of sAHP and self-excitation currents. The input is applied to three TCs with varying resting membrane potentials and self-excitation synaptic currents. The elevated plateaus observed in the curves above the red lines indicate the depolarization of the membrane potential caused by the input signal. However, despite the depolarization, the cells are unable to spike as their membrane potentials remain below the firing threshold due to the hyperpolarizing effect of the sAHP current, which makes the membrane potential more negative.

Fig. 3.

The impact of varied resting potentials on neuronal dynamics: Time cells (TCs) exhibit distinct responses to square pulses, requiring sufficient decay of the sAHP current before firing. The duration of decay is determined by the resting membrane potential of each TC, with higher potentials resulting in shorter decay intervals. Once a cell spikes, self-excitation helps sustain its spiking activity, which ceases when the sAHP current reactivates

Since the sAHP current decays slowly, it takes time for the negative influence of the sAHP current on the membrane potential to diminish. Cells with lower resting membrane potentials require more time for their membrane potentials to be positively affected by the decay of the sAHP current (depolarization due to the decay of sAHP current). As a result, the delay to the first cell’s spike is shorter than that of the second cell, and similarly, the delay to the second cell’s spike is shorter than that of the third cell. These delays in spiking times arise from the variations in resting membrane potentials.

The simulated cells have resting membrane potentials of -67mV, -69mV, and -71mV, respectively. Even slight disparities in resting membrane potentials can lead to significant delays in spiking times of the cells. The sAHP current, which has slow inactivation kinetics, hyperpolarizes the cells and gradually inactivates with the onset of external input. Once the sAHP current sufficiently deactivates and the cells become depolarized, they can start firing. It is crucial to note that the delay in firing time does not progress linearly with a decrease in resting membrane potential. This nonlinearity arises from the exponential inactivation of the sAHP current. Thus, even a slight decay in resting membrane potential can result in substantially longer delays in the firing times of the cells.

To maintain the firing activity, self-excitation plays a crucial role by delaying the effects of adaptation. The sAHP current gradually accumulates during ongoing spiking activity. Without self-excitation, the cells would stop firing much more quickly. The strength of self-excitation determines the duration of firing activity, with stronger excitation leading to longer periods of activity. Thus, once a cell initiates spiking, it can continue to spike with the assistance of self-excitation. On the other hand, self-excitation at the same time also facilitates the build-up of the sAHP current. When the spiking stops due to the sufficient activation of the sAHP current, the cell will not be able to fire for an extended period of time.

The spike windows of cells with higher resting potentials are significantly shorter compared to those with lower resting potentials. This is because for cells with lower resting potentials, the sAHP current requires a longer decay time to sufficiently depolarize the cells. However, in the later stages of the decay, the cells remain closer to their firing threshold for a longer duration, making them more susceptible to the effects of noise. Hence, the longer time interval required to depolarize the cells increases their vulnerability to noise-induced fluctuations in activity (Fig. 4). This observation aligns with the concept of time-scale invariance. Therefore, the main contribution of this work is the demonstration that the interplay between the sAHP current and variations in resting membrane potentials can account for the increased variability in spike times of time cells as time intervals progress.

Fig. 4.

Example ensemble behavior: The figure illustrates the simulation of three time cells (TCs) with distinct resting membrane potentials across 30 trials. The plot displays the average spike latencies along with the corresponding standard deviations represented by red error bars. The variability in spike timing, indicated by the increasing standard deviations, is influenced by the differences in the resting membrane potentials of the TCs

Larger Network Oscillations and Statistical Results

A network simulation was conducted with 50 neurons that incorporate sAHP and self-excitation currents (Fig. 5). It was assumed that the resting membrane potentials of the neurons followed a uniform distribution ranging from -63 to -73mV. The sAHP currents were activated at the beginning of the simulation. The neurons were ordered based on their resting membrane potential levels, with the neuron having the highest potential being the first to fire. The observed behavior in the simulation was characterized by a turn-taking pattern in the firing of the neurons. Neurons with higher resting membrane potentials fired first, as they required less depolarization with the decay of the sAHP current. The self-excitation mechanism in the network leads to the reactivation of the sAHP current in active neurons. As a consequence, after a neuron stops firing, it remains silent for a significant duration. Reactivation of the sAHP current contributes to the extended period of inactivity in the neuron, preventing it from firing again immediately.

Fig. 5.

Raster plot of the larger network simulation: Simulation of 50 time cells (TCs) arranged in order of decreasing resting membrane potentials. The firing of TCs follows a temporal pattern determined by their individual resting membrane potentials. As time progresses, the variability among TCs increases, leading to increased imprecision

To examine the variability in the mean spike latencies of the TCs, we performed 30 simulations of the network (Fig. 6). The underlying cause of the increased sensitivity to noise over time can be attributed to the exponential decay of the sAHP current. The exponential decay inherently entails a decreasing rate of change as time progresses. Consequently, if the resting membrane potential is low, it will take a longer duration for the sAHP current to inactivate and depolarize the cell, resulting in the cell remaining closer to the firing threshold for an extended period. As time elapses, the depolarization caused by the decay of the sAHP current becomes less significant, and noise becomes more influential in determining when the cell initiates spiking activity.

Fig. 6.

Statistical results: A network of 50 time cells (TCs) is simulated over 30 trials. The TCs are ordered based on their resting membrane potentials, with the first TC having the highest value. The upper left panel displays the mean spike times of the TCs, with error bars indicating the standard deviations. The upper right panel illustrates the relationship between the standard deviation and the mean spike latencies. Additionally, the coefficients of variation (CV) for the spike times are plotted against the mean spike latencies. Notably, the CV converges to a constant value, demonstrating the scalar property of the network

The variability in spike times increases with the temporal index of neurons. Figure 6 shows the statistical results of the simulations. Notably, the standard deviation of spike times exhibits a linear relationship with the mean spike latencies, leading to the convergence of the coefficient of variation to a constant value. Consequently, the manifestation of time-scale invariance becomes evident through the output of the system.

Dependency of CV to sAHP Decay Rate and the Standard Deviation of the Noise

We investigated the dependence of the coefficient of variation (CV) on the sAHP current decay rate, represented by ${\beta }_s$ and the standard deviation of the noise. As the time cells initiate spiking activity with the decay of the sAHP current, a higher decay rate leads to faster onset of spiking. Consequently, the speed of the resulting temporal pattern of activity is directly influenced by the decay rate.

However, it is important to note that the wave speed does not have any impact on the variation of mean spike latencies of the neurons. Regardless of wave speed, the CV converges to a constant value for different values of the sAHP current decay rate (Fig. 7).

Fig. 7.

Relationship between the coefficient of variation (CV) and the decay rate (${\beta }_s$) of the sAHP current: CV remains constant for different parameter values

Figures 7 and 8 present the simulation results, with 30 simulations conducted for each parameter setting of the sAHP current decay rate ${\beta }_s$. For larger values of ${\beta }_s$, the depolarization caused by the decay of the sAHP current occurs more rapidly, resulting in faster transitions in the temporal patterns of activity. On the contrary, smaller values of ${\beta }_s$ lead to slower transitions.

Fig. 8.

Relationship between the frequency distributions and the decay rate (${\beta }_s$) of the sAHP current: The primary impact of modifying the parameter ${\beta }_s$ becomes apparent in the speed of wave propagation, with higher parameter values resulting in faster progression

Figure 9 presents the simulation results, with 30 simulations conducted for each parameter setting of the standard deviation of the noise (std). As anticipated, variations in the standard deviation of the noise had a significant impact on the spike time variability of the TCs. Networks exposed to higher noise levels exhibited elevated CV values. It is noteworthy that the CV’s tendency to converge to a constant value remained resilient in the face of fluctuations in the standard deviation of the noise.

Fig. 9.

The relationship between the coefficient of variation (CV) and the standard deviation of noise (std): The CV tends to stabilize at a higher constant level as the standard deviation of applied noise increases

Dependency to Initial sAHP Levels: Temporal Scaling

Temporal scaling refers to the adaptive behavior of neurons in response to changes in the target interval (e.g., see De Corte et al., 2022; Wang et al., 2018). This phenomenon becomes evident when one examines the activity profiles of individual neurons. For example, in a study conducted by Mello et al. (2015), recordings were made in the striatum of rats trained in a task in which the target interval varied randomly across blocks of trials within a session. The activity of individual neurons exhibited peak responses at specific times during the experiment. However, when the target interval was modified, the peak timing of these neurons also shifted accordingly. Wang et al. (2018) suggested that this flexible adjustment could be influenced by an external input. To assess the compatibility of our model with this observed temporal scaling, we performed simulations with varying initial levels of sAHP activation. In Fig. 10, we present the frequency distribution of TC15 and TC20 based on the initial activation levels of the sAHP current. As the initial levels of sAHP activation increased, the neurons required longer periods of waiting for the adaptation current to deactivate before firing.

Fig. 10.

Frequency distributions of TC15 (left panel) and TC20 (right panel) are shown for various initial values of the sAHP activation variable $m_0$.: The peak firing times of the cells exhibit an increase based on the initial activation values of the sAHP current

Comparing the Model Results to Medial Prefrontal Cortex (mPFC) Data

To assess the temporal precision and other properties of our model, we replicated certain aspects of the study conducted by Tiganj et al. (2017) in their analysis of in vivo data from time cells in the medial prefrontal cortex (mPFC). By doing so, we aimed to establish a direct comparison between our model predictions and the experimental observations of time cells in vivo.

Similar to the methodology used by Tiganj et al. (2017), we calculated the instantaneous frequencies of all time cells by tallying the number of spikes within fixed time bins (1000 ms). The resulting raster plot and average frequency distribution curves for different time cells are presented in Fig. 11. The frequency distribution curves of the time cells exhibit peak values at distinct time points within the delay interval, contingent upon the subthreshold potential levels. Notably, time cells characterized by higher subthreshold potentials display peak frequencies occurring at earlier time points compared to those with lower subthreshold potentials. The red curves overlaying the frequency distributions represent Gaussian fits, providing an estimate of the firing field widths based on the standard deviation of the Gaussian curves. Importantly, as time progresses, the width of the firing fields, as indicated by the standard deviation, expands.

Fig. 11.

Raster plots and frequency distributions of individual time cells: Each dot represents a spike from the respective neuron in the raster plots. The cells are arranged based on their subthreshold potential level, which also determines the timing of peak firing

To assess the temporal accuracy as time progresses, we employed the concept of ensemble similarity for time bins across all time cells, following the approach described by Tiganj et al. (2017). This involved calculating the cosine of the angle between vectors representing each time bin. Each time bin vector captures the normalized mean frequency information of each time cell associated with that specific time bin. The resulting heatmap, depicted in Fig. 12, portrays the ensemble similarity. Regions displayed in dark red indicate the highest correlation between bins, while blue colors signify decreased correlations. As anticipated, the diagonal line of the heatmap appears as a dark red color, representing the within-bin similarity of normalized frequencies. Moving away from the diagonal line, the similarity gradually diminishes. Notably, during the later stages of the delay period, the similarity extends further compared to the initial stages of the activity. Similar conclusions can be drawn from the heatmap illustrating the normalized frequencies of time cells, as depicted in the second panel of Fig. 12.

Fig. 12.

Heat maps depicting the similarity between activity time bins (left panel) and normalized frequencies (right panel): Each pixel represents the similarity between the corresponding time bins. The diagonal line indicates self-similarity and is represented by a dark red color. In the later stages of the delay period activity, the similarity becomes more spread out, indicating a decrease in temporal precision. This is further supported by the dispersed activities of cells that peak in the later stages of the delay activity

In the final step of comparing our model results to those of Tiganj et al. (2017), we generated a dot-plot illustrating the standard deviations of Gaussian-shaped frequency estimates, along with the cumulative distribution function (CDF) of peak firing times, as presented in Fig. 13. Consistent with the experimental findings, we observe that the width of the time fields increases as the time of peak firing progresses. This provides further evidence of the growing dispersion of time fields over time. Additionally, the non-uniform distribution of peak firing times indicates that later time points are represented by a smaller number of time cells compared to earlier time points.

Fig. 13.

Standard deviations (left panel) and cumulative distribution function (CDF) of peak times (right panel): Each dot represents a time cell. In the later stages of the delay period activity, there is an increase in the standard deviation, indicating a decrease in temporal precision (left panel). The distribution of peak activity times is non-uniform, with fewer time cells representing later times in the delay period activity

Comparing Model Results to Data from a Peak-Interval Procedure Experiment

The peak interval procedure involves training animals (particularly rodents) to perform a behavioral task that requires them to estimate time-specific intervals of time. During the peak interval procedure, animals are typically presented with a cue or a signal that indicates the beginning of a timed interval. They are then required to wait for a specific duration of time, known as the target interval, before responding or performing a specific action. In order to assess the agreement between our model results and experimental findings, we conducted a comparison by examining the normalized frequency distribution of individual neurons. To obtain reference data, we utilized the peak interval procedure conducted by Swanton and Matell (2011) with rats. In Fig. 14, we present the overlapping plots of the normalized frequency distributions obtained from TC15 and TC20 of our model with the peak interval data derived from Swanton and Matell (2011)’s study, specifically for intervals of 4 s and 12 s. Notably, upon normalization, the model data exhibits a remarkable correspondence and close alignment with the experimental data. This suggests a strong agreement and accurate replication of the observed patterns between our model and the experimental findings.

Fig. 14.

Comparison between the firing rates derived from the model data and the experimental data obtained through the peak interval procedure: The normalized data from the peak procedure experiment (Swanton & Matell , 2011) and the model are superimposed for comparison. The normalization procedure involves adjusting the data relative to each subject’s maximum rate and peak time. The experimental data are obtained from trials with intervals of 4 s and 12 s. Regarding the model data, it corresponds to TC15 and TC20 with average firing times of 15 s and 25 s, respectively

The Interval Bisection Model and Its Comparison with Experimental Findings

To provide further validation for the model’s performance, we introduced response units (RU), specifically RU1 and RU2, to illustrate the model’s behavior in the interval bisection task (see Fig. 15). In this task, an animal initially learns two reference time intervals and is tasked with categorizing test durations as either short or long. The results are quantified by the proportion of choices favoring the “long” category, forming what is referred to as the choice function.

Fig. 15.

Interval bisection network overview: The TC network is augmented with left and right response units (RU) representing short and long reference durations, facilitating the execution of the interval bisection experiment. Training for reference durations is accomplished through the strengthening of excitatory synapses connecting TCs and RU neurons. The decision-making process is governed by lateral inhibition among the RU neurons

Analysis of the experimental data reveals that this choice function follows an S-shaped sigmoidal pattern, with the point of subjective equality closely aligning with the geometric mean of the short and long reference durations. Additionally, when these choice functions are plotted against logarithmically spaced ordered points, they exhibit significant overlap (e.g., see Akdoğan & Balcı, 2016; Balcı & Simen, 2014; Church & Deluty, 1977; Zeki & Balcı, 2020).

To assess the model’s performance, we conducted training sessions with the network. TC (time cell) spikes occurring within the 500 ms interval of the short or long reference duration were reinforced for 300 trials, depending on the respective reference duration. The model’s performance is assessed using seven logarithmically spaced points across interval pairs of 4 s-16 s, 5 s-20 s, and 6 s-24 s. During probe trials, the decision-making process relied on lateral inhibition among the RU neurons.

The model’s choice behaviors are illustrated in Fig. 16a, and they are compared with the corresponding experimental data extracted from the rat bisection dataset (Crystal et al., 2003). Notably, the choice functions closely resemble the S-shaped curves observed in the experimental data, with the point of subjective equality aligning with the geometric mean of the reference durations. Importantly, when these choice functions are superimposed onto the logarithmically spaced points, they exhibit remarkable overlap, further supporting the model’s accuracy and fidelity to the experimental findings (Fig. 16b).

Fig. 16.

Choice patterns in the interval bisection network: The proportion of long responses at logarithmically spaced time intervals is presented for both the model’s results (4 s-16 s, 5 s-20 s, and 6 s-24 s) and the experimental data (2 s-8 s, and 4 s-16 s). The choice functions exhibit S-shaped curves, with the point of subjective equality (PSE) closely aligning with the geometric mean of the short and long reference durations (upper panel). Remarkably, when superimposed, the choice functions from both the model and the experimental data exhibit significant overlap (lower panel)

Discussion

We proposed a neurobiological model for the hippocampal time cells, in which these cells are postulated to possess an intrinsic current known as sAHP, in addition to self-exciting synaptic connections. We did not make any other assumptions about the network architecture. It is assumed that the cells exhibit a heightened activation of sAHP when the external stimulus begins. This heightened activation can be achieved by providing a strong input to the network at the start of the trial. The striatal beat frequency model also incorporates a comparable resetting mechanism (e.g., see Matell et al., 2004; Oprisan & Buhusi, 2014).

Our model exhibits similarities to those proposed by Tiganj et al. (2015) and Killeen and Grondin (2022) in terms of involving an exponential decay mechanism for decoding timed events. For example, Tiganj et al. (2015) employed cells with slowly decaying firing rates to encode time using persistently firing neurons. In Killeen and Grondin (2022), an exponentially decaying function was directly used to mark the epochs of timed experiments. The main distinction in our work lies in its simplicity, as we do not require multiple intrinsic and synaptic currents to model the characteristics of interval timing data.

The proposed model relies on two crucial assumptions concerning the intrinsic and synaptic mechanisms. Firstly, we assume that the participating neurons exhibit self-excitation, which can be observed as feedback excitation within a cluster of excitatory cells. The second assumption is that the initial activation of sAHP in the time cells is nearly identical. This uniform activation can be achieved by providing a strong input to the time cells at the beginning of the experiment. Indeed, similar resetting assumptions have been employed in other interval timing models, as seen in studies such as Simen et al. (2011), Oprisan and Buhusi (2014), and Gibbon et al. (1984).

The functions of the sAHP current in our model can be divided into two components: (1) delaying the spike and (2) facilitating the development of adaptation. It is worth noting that other intrinsic currents can also contribute to spike delay. Specifically, hyperpolarizing currents that are active during the resting state and gradually deactivate with the depolarizing input can fulfill the initial role of the sAHP current. An example of such a current is the D-type potassium current. Regarding adaptation, a calcium-dependent potassium current can serve as the required mechanism. However, the rapid recovery of this current may still necessitate the presence of the sAHP current in the model.

While the wave-like behavior of time cells in the hippocampus suggests the presence of a specific chain-like network architecture, our study demonstrates that temporal patterning can also occur without relying heavily on assumptions about network architecture. We propose that slow dynamics, such as the sAHP current investigated in this work, may account for the observed temporal patterns. The key contribution of our work is to illustrate that small fluctuations in membrane potential, in conjunction with the sAHP current, can elucidate the behavior of hippocampal time cells. These small membrane potential variations could arise from variances in the number of synaptic connections received by the cells. Additionally, the assumptions made regarding resting membrane potentials can be substituted with discrepancies in the spiking thresholds of individual neurons. In this case, the primary source of variations may be attributed to intrinsic current fluctuations within the individual neurons.

In an effort to maintain simplicity in our model, we employed integrate-and-fire type neurons, a choice that significantly reduces model complexity. Furthermore, we refrained from introducing additional assumptions about the network architecture. Specifically, we found that assuming random connections among TCs did not yield any noticeable advantages for the model; instead, it primarily added noise to the spike timing of the neurons. As a result, we opted to keep the model’s architectural structure unrestricted, allowing for greater flexibility while preserving a minimalist approach.

To validate our model and compare its predictions to experimental data, we reproduced some analyses performed by Tiganj et al. (2017) on time cells recorded from the medial prefrontal cortex (mPFC). We observed similar patterns in terms of peak frequencies, firing field widths, and distribution of peak firing times. These findings support the notion that our model captures essential aspects of time cell dynamics and provides a meaningful framework for understanding their behavior. Furthermore, we compared normalized spike frequency data with the experimental findings from the peak procedure experiment done in Swanton and Matell (2011). When superimposed, the normalized frequency data for different time cells overlaps with the experimental data obtained for 4 s and 12 s peak interval procedures.

Author Contributions

Both authors contributed to the development of the research idea and the design of the study. MZ was primarily responsible for the analysis. MZ took the lead in writing the initial draft of the manuscript, while FB provided critical feedback and contributed to the revision process. FB also provided overall supervision and guidance throughout the project. All authors have read and approved the final version of the manuscript.

Data Availibility Statement

The data used in this study are secondary data obtained from external sources. Access to the original dataset is subject to the data usage and sharing policies of the respective sources. For data access, please refer to the following sources: Peak procedure: https://timingdatabase.shinyapps.io/DownloadUI Temporal bisection: Jonathon Crystal, jcrystal@indiana.edu The data utilized in this research are not owned by the authors. For data access, please contact the original data source(s) directly.

Declarations

Ethical Approval

Not applicable for this study.

Consent to Participate

Not applicable for this study.

Competing Interests

The authors declare no competing interests.