Temporal integration on the dendrites of fast-spiking basket cells

Abstract

Neurons receive synaptic inputs with diverse temporal patterns in vivo, and their integration of these patterns is critical for understanding information processing mechanisms in the brain. Fast-spiking basket cells, which perform both supralinear and sublinear dendritic integration, are essential for inhibitory control in the hippocampus. However, their responses and the mechanisms underlying different temporal input patterns remain unclear. To address this question, we apply inputs with varying windows of time to a detailed compartmental model of basket cells. Our results reveal that when synaptic inputs are randomly dispersed, temporal integration in FS BCs exhibits a sigmoid-like response within the temporal window. In contrast, synchronous input protocols more effectively elicit action potentials, while asynchronous inputs generate more spikes in response to suprathreshold stimuli. Further analysis shows that the supralinear dendrites of fast-spiking basket cells primarily mediate this nonlinearity to asynchronous inputs, owing to their larger dendritic diameters. Moreover, we discover that delayed rectifier channels reduce sensitivity to synchronous inputs, whereas N-type channels enhance sensitivity to asynchronous inputs. These results provide insights into the mechanisms underlying the temporal coding of fast-spiking basket cells, which is crucial for understanding their role in neuronal oscillations.

Introduction

Parvalbumin-expressing interneurons account for a minor proportion of the hippocampus but are essential to the synchronizing firing of pyramidal neurons^1,2. As a member of the PV-positive interneuron class, fast-spiking basket cells (FS BCs) are critical for generating oscillations and associated cognitive functions^3–5. Unlike previous neurons that exhibit unitary dendrite integration properties, an intriguing finding shows that the FS BCs in the CA3 area show both supralinear and sublinear integration⁶. Given the intrinsic association between dendritic nonlinearity and the resulting somatic output, investigating the input-output relationship in FS BCs becomes necessary.

The mechanisms by which dendrites integrate synaptic inputs that are temporally or spatially complex have significant implications for neural computation^7,8. Presynaptic inputs have randomness due to the time-dependent character of dendritic conductances, thus modulating the degree of dendritic compartmentalization^9,10. Therefore, nonlinear dendritic integration properties regulate the various responses resulting from different spatiotemporal arrangements of synaptic inputs¹¹. For example, in layer V and CA1 pyramidal neurons, synchronous and clustered synaptic inputs lead to nonlinear integration in dendrites, while asynchronous inputs result in linear integration^12,13. This diversity shapes somatic output and can further modulate network activity. Previous studies on the spatial integration of FS BCs have found that they prefer clustered synaptic inputs rather than dispersed, mainly caused by dendritic morphology and calcium-permeable AMPA receptors¹⁴. However, the mechanism underlying the temporal integration of synaptic inputs in FS BCs remains unclear.

To address this question, we use a multi-compartmental model of FS BCs in the hippocampus CA3 region of the hippocampus. To mimic different degrees of input synchronization, we introduce an equal number of synaptic inputs within temporal windows of 1 ms, 10 ms, 50 ms, and 100 ms, respectively. We find that membrane potential changes in a sigmoid-like manner, and this process is dominated by supralinear dendrites in FS BCs. By manipulating biophysical mechanisms in dendrites, we discover that delayed rectifier channels are more influential in the input sensitivity of synchronous inputs, and N-type channels are more influential for asynchronous inputs.

Results

Response of FS BCs to different temporal synaptic input patterns

We construct a biophysical model of fast-spiking basket cells using realistic morphology from rat hippocampal areas (CA3)¹⁴. As shown in Fig. 1A, the black region represents dendrites, the grey region represents axons, and the “v” symbolizes the soma¹⁵. Basic membrane properties, ion channel conductances, and synaptic inputs are selected based on experimental data^16–21 (Supplementary Tables S1–S3). After injecting hyperpolarizing and depolarizing current with different amplitudes in soma, realistic responses and electrophysiology properties ensure the biological relevance of the model (Supplementary Fig. S1, Table S5).

Figure 1.

(A) Fast-spiking basket cell model and temporal simulation protocol. Dark regions represent dendritic compartments, the axon is shown in grey, and the soma is marked at the location of V. N synaptic inputs (red circles) are generated within the time window from 0.1 to T and placed at the midpoint of randomly selected dendrites (where N is the number of synaptic inputs, and T is the duration of the input window). Each simulation consists of ten cycles, with stimulated dendrites reselected for each cycle. The output is the membrane potential recorded at the soma. (B) Mean input-output relationships for temporal input patterns in somatic recordings: 1 ms (blue), 10 ms (green), 50 ms (yellow), and 100 ms (purple). Data points of the same color represent individual measurements, and all values are shown as means. Each curve is fitted with a sigmoid-like function of the form (Confidence interval = 95%). The shaded regions indicate the corresponding standard deviations. Fitted function: 1 ms (blue): ; 10 ms (green): ; 50 ms (yellow): ; 100 ms (purple): (the value of d is unified to facilitate comparison across conditions).

To investigate the neuronal response to different temporal input patterns, we activate increasing synaptic pairs (ranging from 2 to 30 with increments 2), each containing one calcium-permeable AMPA (cp-AMPA) input and one NMDA input. For each simulation, we randomly select the dendritic branches to mimic the inputs generated by a population of neurons. This procedure ensures that all synapses are dispersed throughout the dendritic tree, excluding branches that fail to meet the specified criteria (Supplementary Table S4). Each simulation repeats ten times with different seeds to encompass a broad range of dendritic branches. Within the specified time windows T (T = 1, 10, 50, and 100 ms), presynaptic inputs to postsynaptic receptors assign random delivery times according to a uniform distribution (Fig. 1A). Selecting suitable dendritic locations for analysis is challenging due to the dispersion of synaptic inputs, and we aim to investigate the role of dendrites in transforming inputs into outputs. To this end, we record somatic membrane potentials for further analysis.

Following the simulations, we observe a progressive somatic depolarization within each temporal window, which increases with the number of synaptic inputs (Fig. 1B, dots). We employ a sigmoid-like function to provide a more intuitive visualization of the curve’s trajectory to fit the data. The function used is y=a/(1+exp(-(x+b)/c))+d, where a is the amplitude, b denotes the midpoint of the rise, c represents the slope and also refers to input sensitivity, and d is a constant to establish the y-axis position of the curve during each fitting²².

The fitted curves reach their maximum when generating stable somatic action potentials. As shown in Fig. 1B, the changes curves of 1 ms and 10 ms temporal windows show a notable similarity (blue and green lines). Meanwhile, except for the 100 ms temporal window, the input windows within the 50 ms exhibit a quite similar peak position.

Previous research in pyramidal neurons has demonstrated a correlation between the number of action potentials (APs) and dendritic integration mode. To investigate this relationship in FS BCs, we analyze the input efficacy, i.e., the accurate threshold points and the number of APs across different temporal windows. Firstly, we find that the number of synaptic inputs required to generate a stable action potential is lower for synchronous inputs than for asynchronous input patterns (Fig. 2A,B). This threshold parameter correlates with the b-value of the fitting curve in Fig. 1B (Pearson correlation coefficient = 0.9649, p < 0.05). Secondly, we investigate the subsequent effect of increasing the input above the threshold on AP output, particularly for asynchronous inputs. As expected, the synchronous input maintains stability within a range twice the threshold number of inputs. Similar to the changes in membrane potential, this stable characteristic can be sustained within the 10 ms input window. Conversely, as the number of synaptic inputs surpasses the threshold, the increase in APs is proportional to the degree of asynchrony (Fig. 2C).

Figure 2.

(A) Plot of the required number of synapses for spike generation across four temporal simulation protocols. The threshold is the minimum number of synaptic inputs necessary to generate a single spike consistently across all cycles. The y-axis indicates the number of synapses, with inputs arranged in pairs. Error bars represent the standard deviation. (B) Somatic membrane potential recordings show the required number of inputs to generate a single spike under different time windows. The legend indicates that higher levels of synchronous inputs reduce the inputs needed to trigger a spike. The grey dotted line marks 0 mV. (C) Plot of the number of spikes evoked as a function of input size relative to the threshold. Non-integer values may appear since the results represent average values across multiple cycles.

The above simulations reveal that the input-output curve of dendritic temporal integration in FS BCs exhibits similar nonlinearity over longer time windows. Meanwhile, input efficacy decreases with increasing temporal windows, whereas the number of action potentials is more likely to increase when synaptic inputs surpass the threshold.

Supralinear dendrites with larger dendritic diameters dominate excitatory responses to the asynchronous inputs in FS BCs

The temporal summation of excitatory postsynaptic potentials (EPSPs) at the soma is shaped by dendritic geometry and relevant input resistance²³. Previous studies have shown that the dendrites of FS BCs can perform bi-modal dendritic integration. This characteristic allows them to perform supralinear and sublinear integration on the same dendrite, mainly determined by their morphology⁶. Meanwhile, the observed saturation in the membrane potential response can be attributed to generating dendritic spikes in supralinear dendrites. Therefore, our investigation delves into the type of dendrite integration that determines the temporal summation in these cells.

Firstly, we separate supralinear and sublinear dendritic compartments by comparing the recorded somatic EPSPs with linearly summed EPSPs. A dendrite is classified as supralinear if the measured EPSPs exceed the expected values (linearly summed EPSPs), while it is categorized as sublinear otherwise (Supplementary Table S4, Fig. S3)^14,24. Following this classification, we apply identical stimulus protocols to supralinear and sublinear dendrites, respectively.

After conducting the simulations, we observe a noticeable discrepancy in the response trends between supralinear and sublinear dendritic sections, even within the same number of inputs. We employ the SmoothinSpline Method in Matlab to effectively smooth these curves, as some data points do not conform well to a sigmoid function. Compared to Fig. 1C, the trajectories for supralinear dendrites remain consistent, especially for more synchronous inputs (Fig. 3A,C). Conversely, when stimulated solely on sublinear dendrites (Fig. 3B), the amplitude varies within a narrow range. These variations are insufficient to induce any sigmoid-type nonlinear shift under the same input conditions. Therefore, we hypothesize that supralinear dendrites dominate the temporal integration efficiency of FS BCs.

Figure 3.

(A) Mean input-output relationships for supralinear dendrites only. The Smoothing Parameter for each curve is set to 0.011412. Shaded regions represent the corresponding standard deviation. Sigmoid-fitted functions are provided in Supplementary Table S6. (B) Mean input-output relationships for sublinear dendrites only. (C) Comparison charts for the entire dendritic tree (grey), supralinear dendrites (red), and sublinear dendrites (yellow) across four different time windows. The grey curve, used as a reference, is re-fitted using the SmoothinSpline Method.

Given the restricted number of sublinear dendrites, simulations on them may have spatial constraints. To further investigate the role of supralinear dendrites, we selectively block sodium channels in the dendrites, eliminating supralinearity in FS BCs⁶. Since all dendrites are depicted as sublinear, we conduct simulations across the whole dendritic tree without distinguishing regions. After the simulations, we observe delayed sensitivity in the nonlinear summation (Fig. 4A). Further investigation into input thresholds provides additional evidence for this point, as the increased threshold is proportional to the length of the temporal window (Pearson correlation coefficient = 0.9883, p < 0.05) (Fig. 4B). These findings suggest that supralinear dendrites with dendritic spikes enhance the sensitivity of temporal integration, particularly to asynchronous inputs.

Figure 4.

(A) Mean input-output relationships after blocking dendritic channels. All curves are fitted using the SmoothinSpline Method. (B) The required threshold to generate a single action potential after blocking sodium channels in the dendrite (left y-axis). The difference in the threshold between control conditions and sodium channel blockade (right y-axis).

Blocking sodium channels changes the dendritic integration type by altering active properties in neurons. As mentioned above, dendritic diameter is another decisive factor in bi-modal dendritic integration as passive membrane properties⁶. To demonstrate the functional importance of supralinear dendrites, we adjusted the dendritic diameter to 0.2 μm, a relatively small dimension even for sublinear dendrites. Paralleling the effects of sodium channel blockade, reducing the dendritic diameter also attenuates the rise in membrane potential to asynchronous input, particularly evident at 50 ms (Fig. 5). In this state, neurons are unable to generate action potentials within each cycle reliably.

Figure 5.

Mean input-output relationships after adjusting the dendritic diameter to 0.2m. All curves are re-fitted using the SmoothinSpline Method with the same smoothing parameter as above.

These findings suggest that supralinear dendrites facilitate the temporal integration of inputs in FS BCs, primarily due to the dendritic sodium channels and their large dendritic diameters. This observation also illustrates from another perspective that supralinear—but not sublinear—dendrites generate dendritic spikes that further boost the induced EPSPs²⁵.

channels play a crucial role in regulating the temporal input summation of FS BCs

In addition to sodium ion channels that can modulate dendritic integration properties, neurons are endowed with various ion channels within their dendrites. Previous studies have shown that dendritic potassium channels can shape the time course of EPSP in FS BCs²⁶. To investigate whether dendritic channels influence the temporal summation, we repeat the above simulations while blocking the expression of Kv3 channels. These channels are one type of delayed rectifier channel and a hallmark of FS BCs^27,28. When the delayed rectifier potassium channels are inactivated in distal and proximal dendrites, the somatic depolarization shows that this conductance significantly impacts the input-output relationships across all temporal stimulation protocols. After fitting the data with the sigmoid-like equation , we observe the differences among various time windows become less pronounced, indicating a diminished sensitivity to the length of the input window during this state (Fig. 6A). Similar changes are noted when the conductance of delayed rectifier channels () decreases by halving its values (Fig. 6B). However, upon comparing the three scenarios, we find the impact of varies across different time windows (Fig. 6C).

Figure 6.

(A) Mean input-output relationships after setting the delayed rectifier conductance to zero. Fitted funtion: 1 ms (blue): , 10 ms (green): , 50 ms (yellow): , 100 ms (purple): . A significant similarity between the 1 ms and 10 ms data results in a substantial overlap of their respective curves. (B) Mean input-output relationships when the delayed rectifier conductance is reduced by half. Specific sigmoid-fitted functions are presented in Supplementary Table S7. (C) Comparison chart across four time windows, with delayed rectifier conductance set to normal (grey), reduced by half (red) and set to zero (yellow).

For a more intuitive comparison, we analyzed the variations in parameter b (normal- blockade) and the ratio of parameter c (normal/ blockade) in the fitted curve (Fig. 7). Given the positive correlation between the parameter b and the firing threshold, the results reveal that the threshold reduction mediated by is proportional to the widening of the input window (Pearson correlation coefficient = 0.9859) (Fig. 7A). This correlation shows that, like the facilitation of sodium channels, the inhibition of temporal integration by potassium channels is time-dependent. Meanwhile, the parameter c, which reflects the sensitivity of neuronal response, is another noteworthy parameter to consider. By comparing the ratios across four temporal windows, it is observed that blocking enhances input sensitivity under more synchronous temporal inputs (1 ms and 10 ms) while exhibiting minor impact on sensitivity for more asynchronous inputs (Fig. 7B, green).

Figure 7.

(A) Parameter b from the fitting function in Fig. 6A (left y-axis). The difference in parameter b between the normal condition and the condition after blocking the delayed rectifier conductance (right y-axis). (B) Parameter c from the fitted function as shown in Fig. 6A (left y-axis). The ratio of parameter c under normal conditions to that observed after blocking the delayed rectifier conductance (right y-axis).

In summary, the conductance of delayed rectifier channels affects the response threshold across all temporal input protocols, particularly for asynchronous inputs. However, potassium channels significantly reduce the sensitivity of synchronous inputs to changes in synaptic input number. This reduction occurs because blocking dendritic channels alters the decay time course of unitary EPSPs, prolonging the time window for temporal summation and accelerating the activation of other ion channels under synchronous inputs²⁶. Moreover, given the critical role of channels in promoting membrane hyperpolarization, any reduction or complete blockade of conductance increases the depolarized amplitude²⁹, consistent with the increase in parameter a.

N-type channels play a crucial role in regulating the asynchronous input summation of FS BCs

The study above elucidated the varying influences of sodium and potassium ion channels in temporal integration. However, an intriguing observation from Fig. 1B is that, while the 50 ms stimulation window does not require the fewest synapses for membrane potential increase, it exhibits the steepest rate of rise. Among the ion channels that contribute to the depolarization of the membrane potential, the N-type channels stand out due to their relatively slow activation and deactivation kinetics^30,31. Therefore, we are interested in whether this characteristic affects the sensitivity to asynchronous input.

After blocking the conductance of N-type channels(), we observe that the overall trend remains relatively consistent, but the extent of change varies across different windows (Fig. 8A). A comparative analysis of the equations and curves reveals that has negligible impact on the 1 ms and 10 ms time windows. However, as the temporal window expands, the rising rate of the membrane potential decreases, especially within the 50 ms temporal windows (Fig. 8B). The ratio of parameter c (normal/ blockade) in the fitted curve underscores this observation, highlighting the nuanced role of N-type channels in modulating the neuronal response to asynchronous synaptic inputs over different time scales (Fig. 8C).

Figure 8.

(A) Mean input-output relationships when the N-type calcium conductance in dendrites is blocked. Fitted funtion: 1 ms: , 10 ms: , 50 ms: , 100 ms: . (B) Comparison chart across four-time windows before (grey) and after (red) the blockade of . (C) The ratio of Parameter c under normal conditions compared to that after the blockade of the conductance.

These data suggest that N-type calcium channels in dendrites enhance the sensitivity to synapse numbers during asynchronous responses while having no discernible effect on more synchronized inputs.

Discussion

Fast-spiking basket cells exhibit precise temporal responsiveness to excitatory synaptic inputs in vitro^32,33. However, the temporal patterns of synaptic inputs in vivo exhibit considerable heterogeneity³⁴. The complex temporal stimuli challenge understanding the response in FS BCs to varying degrees of asynchronous input and the mechanisms underlying these responses. To address this knowledge gap, we simulate distinct temporal asynchrony protocols on a compartmental model of basket cells to elucidate their response mechanisms to inputs of diverse temporal structures.

Our results reveal that when synaptic inputs are randomly dispersed, temporal integration in FS BCs follows a sigmoid-like response within the temporal window. This pattern, observable through the curve and fitted functions, indicates that FS BCs show limited discrimination in their responses within a 10 ms temporal window (Fig. 1). In addition to accurately detecting incoming inputs, the reliable transformation of these inputs into precise temporal outputs across different time windows is also crucial for neurons to convey information precisely³⁵. Further analysis reveals that synchronous inputs are more likely to elicit action potentials, indicating a more robust response to these inputs. In contrast, the suprathreshold inputs under asynchronous inputs lead to an increasing number of spikes (Fig. 2). The cable theory may explain the above results: asynchronous inputs contain more low-frequency components than synchronous inputs, which suffer less signal attenuation and thus induce more spikes³⁶. Additionally, dendrites processing signals nonlinearly can functionally detach as independent nonlinear computation units³⁷. This feature allows neurons to respond flexibly based on input quantity and specific spatiotemporal patterns¹⁶. Therefore, the robustness of nonlinear dendritic processing in FS BCs under asynchronous input patterns may provide insights into the intrinsic mechanisms underlying elevated firing rates.

Unlike “on-off” cells, the dendrites of FS BCs exhibit a unique capability to integrate inputs in both supralinear and sublinear manners⁶. The results under separated simulations show the predominance of supralinear dendrites in the temporal summation of FS BCs (Fig. 3). This result can linked to the presence of sodium channels and the lower input resistance corresponding to the large dendritic diameter of these supralinear dendrites (Figs. 4, 5). These properties enhance the recruitment of dendritic electrogenesis, thus facilitating more significant excitation to the soma³⁸. Moreover, given that the electronic properties enable dendrites to perform nonlinear integration under spatiotemporal activation^39,40, our modulation of channels reveals their essential role in dendritic temporal summation (Fig. 6). Analyzing the fitted function reveals that delayed rectifier channels can lower the input threshold proportionally and predominantly suppress input sensitivity within the synchronization window (Fig. 7). However, the absence of dendritic N-type calcium channels yields a dissimilar regulatory outcome to Na⁺ and channels. Our findings show that reducing dendritic N-type channels attenuates the response rate of asynchronous inputs only (Fig. 8). This effect may be attributed to the slower activation and inactivation kinetics of N-type calcium ion channels^41,42. Moreover, our findings provide another perspective on the dispersed sensitivity of FS BCs, as dendritic spikes can also emerge under the asynchronous inputs, driving the membrane potential towards saturation (Fig. 1B). These observations suggest that FS BCs possess nonlinear integration capacities during both highly synchronized neuronal oscillations (sharp waves and ripples) and within oscillations of lower frequencies^43,44.

Brain rhythms such as theta, gamma, and sharp-wave ripples are essential coordinators of interaction with the hippocampus and other structures involved in learning and memory, with each rhythm playing a unique role^45–50. Perisomatic inhibition mediated by FS BCs is vital for controlling and synchronizing the action potentials of principal neurons across various neuronal oscillations and regulating sensory processes^51–53. Therefore, the temporal integration of FS BCs for inputs within different time windows significantly impacts their function in different brain states. Our findings provide a possible explanation for the temporal summation features and firing properties of FS BCs under various network conditions. This explanation further lays the foundation for the precise timing of inhibitory outputs that determine the initiation of action potentials in principal neurons, ultimately shaping the flow of information⁵⁴.

Although our findings provide insights into temporal integration in FS BCs, several limitations should be acknowledged. While regulating ion channels at different sections, we observe that the model exhibits heightened excitability. For instance, blocking sodium ion channels in the soma and axon does not prevent the membrane potential from rising spike-like, which deviates from the typical responses observed in real neurons in vivo. Similar limitations have been observed in models reported in other studies⁶. After adjusting some parameters, we speculate that large-diameter dendrites in the reconstructed morphology may contribute to this phenomenon, which enhances the transmission of electrical signals. Although these dendrites are excluded from our simulation, some are inevitably positioned on the signal propagation pathways. Given the close correlation between morphological parameters and neuronal properties, altering the morphology for simulations is not straightforward. Additionally, because the locations for the same number of inputs are consistent, and this study focuses primarily on trends rather than absolute values, the conclusions are unlikely to be significantly affected. However, addressing this challenge will remain a key focus of our future modelling efforts.

In addition to receiving excitatory synaptic inputs, FS BCs also receive a minor fraction of inhibitory inputs from other interneurons^55,56. Studies have shown that individual dendrites can process the spatiotemporal association between excitation and inhibition synapses⁵⁷. Therefore, further research is needed to explore the effects of inhibitory inputs and spatial interactions on the temporal summation in FS BCs.

Methods

Modelling

The FS BCs model, derived from our earlier publication¹⁴, includes four sections-soma, axon, proximal dendrites, and distal dendrites-with biophysical properties informed by experimental data. The model incorporates 10 types of voltage-gated ion channels. Three of these channels (fast , delayed rectifier , and A-type channels) are distributed across both the soma and dendrites, four (slow inactivation , fast calcium-dependent slow calcium-dependent , and hyperpolarization-activated current) are assigned exclusively to the soma. The remaining three calcium channels (L, N, and T-type channels) are localized to the dendrites. Every section except the axon implements a calcium buffering mechanism¹⁹. Channel conductance values are provided in Supplementary Table S2⁶. The model is validated against experimental data to ensure biological relevance (Supplementary Table S5).

All inputs used in the simulations are excitatory synaptic inputs, including calcium-permeable AMPA and NMDA conductances, with synaptic weights of 7.5*^17,58 and 3.2**5⁵⁹, respectively. The resting potential of the model is set at −68 mV, and all simulations are performed in the NEURON simulation environment⁶⁰.

Temporal simulation protocols

Synaptic inputs are generated using the NetStim package in NEURON, where each input is modelled as a single depolarizing pulse²⁵. With a fixed number of synaptic inputs, we define the synchrony of these inputs as the time window over which they are transmitted. This time window is set from 0.1 to T, where T = 1 corresponds to synchronous inputs and T > 1 indicates asynchronous stimuli²². All input times within the interval are uniformly distributed.

Dendritic compartments outside the realistic morphological range of FS BCs with an average diameter exceeding 1.2 m are excluded from all simulations⁶¹. To reduce the specificity of the simulation results, we employ random stimuli. Specifically, dendritic branches are randomly selected across the entire dendritic tree or from individual nonlinear dendrites, and each simulation is repeated ten times. All data are recorded at the soma for 200 ms with a time step of 0.1 ms. The reported results represent averages from N cycles (N = 10).

Data analysis

Data analysis is performed using MATLAB (The MathWorks) software. The sigmoid-like functions are fitted using the NonlinearLeastSquares Method in the Curve Fitting Toolbox 3.7. The parameter d in each fit is determined based on the trajectory passing through the initial point (0, − 68) at 1 ms. Other curves are fitted through the SmoothingSpline Method. All values are reported as mean ± SEM (standard error of the mean).

Author contributions

M. L. and XJS designed the study, performed the research, analyzed data, and wrote the paper. All authors agree to be accountable for the content of the work.

Data availability

The source code and the data are publicly available on GitHub (URL: https://github.com/77laundry/temporal-integration-FS-BCs.git).

Declarations

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-81655-w.