We are primarily inspired by the in vitro experiment that the output correlation is determined by the firing rate (de la Rocha et al., 2007). SFA is a prominent characteristic in decreasing the firing rate, which could be modeled by various physiological adaptation mechanisms, such as I_M (voltage-gated K⁺ current; Ermentrout, 1998), I_AHP (C^a2+ gated K⁺ current; Ermentrout, 1998), and I_KNa (Na⁺ activated K⁺ current; Wang et al., 2003). I_M and I_AHP are inhibitory potassium currents that are activated during APs (Benda and Herz, 2003; Benda et al., 2010). These two kinds of currents shape the information transmission properties on a slow time scale (Ha and Cheong, 2017). I_KNa is mobilized over a longer time scale (Wang et al., 2003). In addition, Benda et al. (2010) reproduced the adaptation by dynamic threshold or adaptation current from the perspective of the physical model. In the following part, three biophysical adaptation mechanisms and two adaptation mechanisms generated by the mathematical model are introduced.

The output correlation is generated by correlated input. We choose a pair of neurons receiving fluctuated correlated input current (Barreiro et al., 2012). As shown in Figure 2, the current has a common input component making the pairwise neuron share afferent input. In addition, the neuron pair has an independent input. The fluctuating input current to neuron i (i = 1, 2) follows:

where k = 1, 2, ⋯ , N (N = 1, 000) is the experiment trial and i is the neuron index. μ_i and σi2 are the average and variance of the input current. c (0 ≤ c ≤ 1) is the input correlation coefficient. ξci(t) and ξik(t) represent the common and independent Gaussian white noise process of neuron i in trial k, respectively (Lindner et al., 2005). ξci(t) is the same in different trials, while ξik(t) is randomly generated at each trial. We set appropriate parameters to simulate the variability of Vik (Lampl et al., 1999) and the spike trains generated by Vik (Kohn and Smith, 2005) in vivo condition (Alonso et al., 1996). We adopt Pearson's correlation coefficient ρ (Mark Borodovsky, 2008), which is calculated by:

where Cov and Var represent the covariance and variance. n1 and n2 are the spike counts computed over the time window τ_T. The common input generates correlation between pairwise neurons. The correlation coefficient ρ depends on two factors: time window τ_T and slide window τ_s. ρ is a dimensionless quantity, which ranges from 0 (independent) to 1 (entirely correlated), respectively. The input correlation c bounds the output correlation ρ. The correlation susceptibility S is the slope of ρ to c, which is described by:

where S depends only on the input and output correlation. In addition, the coefficient of variation (CV) is adopted to measure the variation in firing rate and the interspike intervals.

The sequence of time intervals between adjacent action potentials of neuronal firing is called the ISI sequence. The ISI sequence and a histogram of the ISIs in the sequence can inform about both the mode and variability of the firing frequency in neuronal activity and are, therefore, useful measures. All simulations are implemented in MATLAB R2017a using the forward Euler method.

Different adaptation mechanisms are applied to pairwise neurons to examine how SFA shapes the neuronal output response between pairwise neurons. The neuronal output response without the SFA is first explored, as depicted in Figures 3A, B. The firing patterns are sensitive to the correlated input, and the membrane potentials exhibited oscillations. The non-uniform distribution of the patterns resulted from noisy inputs. We then examined the neural activity with adaptation (M current or AHP current). As shown in Figures 3C, D, the M current makes the output responses sparser than those without adaptation. This slowing down of the firing activity also exists in the situation with the AHP current, as shown in Figures 3E, F. The addition of adaptation decreases the firing rate of individual neurons. The inhibitory effect of SFA on slowing the firing patterns might further decrease the output correlation.

The inhibition generated by adaptation slows the spiking activity of single neurons. Next, we investigated how the inhibitory effect changes the output correlation ρ, using the input correlation c, and correlated input intensity μ. The variance of the correlated input is fixed at a constant value. The existence of adaptation made ρ less sensitive to the input correlation, where the slope of the variation slowed down, see Figure 4A. The value of ρ decreased with the addition of adaptation at the same c value. As the value of c ranged from 0 to 0.5 with an increase of 0.05, the output correlation and firing rate are calculated, as shown in Figures 4A, C, respectively. The output correlation for each adaptation is monotonic as c increases. The firing rate increases slowly with the enhancement of c, and the higher conductance intensity induces lower firing rate level. Because of the large ordinate space in Figure 4C, the change in firing rate with c is not obvious, but when viewed at higher resolution (not shown here), firing rate did show a significant increase with c. According to the above experiments, ρ values and the firing rate exhibited similar variation trends as the value of c increased. Therefore, we speculate that adaptation decreases ρ by decreasing the firing rate. We then studied the input intensity's influence on the output correlation ρ. As shown in Figures 4B, D, ρ and the firing rates are both sensitive to μ. The value of ρ increased while μ also increased, and the firing rate showed the same tendency under the same stimulus. The introduction of the M and AHP currents decreased the ρ value, and a decrease in the firing rate accompanied this. Combined with the above speculation of ρ and firing rates, we conclude that the M and AHP currents decrease the output correlation by decreasing the firing rate. The firing rate serves as a bridge connecting the SFA and output correlation, which causes the inhibition of the firing rate at the cellular level and decreases the output correlation at the network level.

To examine how the adaptation current shapes the neuronal output response, we varied the adaptation conductance of the M and AHP currents. The intensity of the adaptation current depends on conductance, which is directly related to the firing rate of the neurons. A stronger adaptation current generates a lower firing rate. As shown in Figure 5A, the correlation decreases significantly with the increase in AHP adaptation conductance compared with the correlation without the SFA. The output correlation slows as the AHP conductance increases. In addition, the enhancement of input enlarges the output correlation generally. Three examples of firing sequences are superimposed with different colors, as listed in Figure 5D. It is obvious that the spike trains of the pairwise neurons become sparse. At gAHP=2.5 mS/cm2, the firing is swift at the beginning. Then the spike train follows with slow sparse spikes. As g_AHP increases to 5 mS/cm² and 10 mS/cm², the rapid firing becomes less noticeable at the initial stage. Furthermore, the spike train correlation becomes weakened. The sparse spiking reflects the slowing firing rate, as shown in Figure 5B. The firing rate tended to decrease as g_AHP increases. The correlation variation corresponds to the firing rate variation, and they present the same decreased variation tendency as g_AHP enhances. Under AHP current, the firing rate variation presents a gentle decreased tendency, which is corresponded to the correlation. This slowing effect changes the v of the output correlation to the input correlation. The susceptibility is displayed in Figure 5C. The results show that the susceptibility decreases as g_AHP increases. The variation trend of susceptibility is consistent with the output correlation. The enhancement of spike frequency adaptation decreases the interaction of pairwise neurons, which further attenuates the susceptibility. Furthermore, the significant difference between different input intensities is investigated to demonstrate the role of input intensity on output correlation, as shown in Figure 5E. The results show that there is a significant difference between different inputs. The external inputs play a pivotal role on output correlation. The enhancement of external input intensity enlarges the correlation apparently.

Furthermore, the role of M current on output correlation is investigated. As shown in Figure 6A, the output correlation attenuates as M conductance increases. During μ = 52 mA/cm², the neuronal firing rate is sensitive to the mean value of the input. However, in terms of μ = 47 mA/cm² and μ = 43mA/cm², the correlation is significantly reduced, beginning with the red curved arrows. The corresponding spike trains are shown in Figure 6D. The firing patterns demonstrate an evident sparse distribution as g_M increases. From gM=0.5mS/cm2 to gM=0.75mS/cm2, the firing patterns present are sparser, which indicate that the firing rate is significantly reduced. As g_M enhances displayed in Figure 6B, the firing rate presents an obvious subdued tendency. The firing rate is relatively low and sensitive to input fluctuations, as shown in Figure 6C. In contrast to the AHP current, although the M current reduced the output correlation, the correlation variation is not smooth. This is because the interplay between the M current and the input intensity slows the firing rate to a minimal value. While for M current, the firing rate generates obvious variations at gM=0.75 mS/cm2 with μ = 47 mA/cm² and gM=0.5 mS/cm2 with μ = 52 mA/cm². These significant variations are corresponding to the correlation variation in Figure 6B. The firing rate variation results in the correlation variation. In addition, the firing rate further decreases the susceptibility of input to output correlation, as displayed in Figure 6C. As shown in Figure 6E, the two-sample t-test indicates that the correlation at the same adaptation conductance under different inputs is significantly difference. The variation of susceptibility is consistent with the trend of output correlation. In conclusion of the role of SFA in output correlation, the correlated input and adaptation conductance determined the firing rate, and the output correlation is further modulated by the firing rate generated by the combined effect. The decreased firing rate depends on the attenuated SFA conductance and further inhibits the output correlation.

In the absence of the SFA, the maximum correlation value generated by the correlated input is approximated by the input correlation, which is consistent with previous findings (de la Rocha et al., 2007; Litwin-Kumar et al., 2011). Prescott et al. (2006) found a terminated firing phenomenon (i.e., the cessation of spiking activity) that causes firing patterns to be generated only at the onset of a stimulus (Prescott and Sejnowski, 2008). In our simulations, this terminated firing phenomenon 'appears' but the noise component of correlated input still generates some sparse firing patterns as depicted in Figure 6D. It is worth mentioning that the form of the terminated firing phenomenon is controlled by two factors: stimulus intensity and adaptation conductance. Low stimulus intensity and adaptation conductance generate the terminated firing phenomenon more easily. This phenomenon causes the sensitivity of the firing rate to change from the input intensity to the noise component of the input, resulting in a significant decrease in the firing patterns, which become very sparse. In conclusion, different SFA mechanisms have various effects on output correlation. However, they control the correlated input or adaptation conductance to change the firing rate and have a decreased impact on output correlation.

Next, we investigated how adaptation affected this correlation. To quantify the different effects of the M current and AHP current, the ISIs and their histograms are calculated. The ISI histograms, under different adaptation conductance, are shown in Figure 7. The ISIs without SFA are distributed at lower values ~10 ms, with a substantial peak. In this case, pairwise neurons displayed a rapid firing process. As the adaptation conductance increases, the adaptation currents gradually increase, decreasing the firing rate. The diagrams with the AHP current in Figure 7A show a regular change: ISIs are distributed over a broader range on the horizontal scale. The values of the ISIs along the x-axis are increasing with increasing adaptation conductance, while the amount of ISIs reduces. The variations in the ISIs in Figure 7B with the M current are similar to those with the AHP current. It is worth noting that there is a significant difference: ISIs with AHP current show a stable change from top to bottom as shown in the subgraph of Figure 7B, whereas ISIs with M current show an evident variation from gM = 0.5mS/cm2 to gM = 0.75mS/cm2 because this adaptation is sufficient to make the firing patterns sparse. The number of ISIs decreases dramatically, and larger values are gradually generated by the noisy component of the input, which becomes a dominant factor in changing the correlation. The large values of ISIs become essential in decreasing the firing rate and further reducing the output correlation when the adaptation conductance increases. As shown in Figures 7C, E, the mean value of the ISIs increased with an increase in the adaptation conductance. The increase in Figure 7C is relatively uniform, whereas the variation trend in Figure 7E slows at gM = 0.5mS/cm2. This is because the AHP current decreases the firing rate slightly, whereas the M current generates a significant decrease in the firing rate at gM = 0.5mS/cm2. Meanwhile, the coefficient of variation of the ISIs at gM = 0.5mS/cm2 increases at the inflection point. For gM>0.5mS/cm2, the firing patterns are generated by correlated noise, which causes the firing patterns to be generated randomly, and the firing becomes more irregular. Under adaptation conductance where gM>0.5mS/cm2, the CV of the ISIs no longer increases but shows a decreasing trend. This trend is consistent with the variation shown in Figure 6A. In conclusion, AHP and M adaptation decrease the output correlation with an increase in the strength of the adaptation currents. The attenuation in output correlation is accompanied by the increment in ISIs distribution.

The dynamic firing process of neurons determines the variation in the special output correlation, while the output correlation depends on the firing rate for a given fixed shared input. There are two stages of firing rate variations in the situation with SFA. From a single neuron point of view, in the first stage, the neurons show rapid firing as shown in the front part of Figure 1G. While in the second stage, the neurons exhibit slow and steady firing. We discovered an interesting phenomenon under the same parameter settings, except for the stimulation time. The experimental results are the opposite. The stimulation times in Figures 8A, B are set to 1,000 and 10,000 ms, respectively. As shown in Figure 8A, the correlation increases because the rapid firing rate in the first stage dominates the correlation. As the stimulation time increases to 10,000 ms, the effect of rapid firing weakens, and the slow firing process in the second stage primarily impacts the correlation. The question then arises as to how long it would take for the stimulation time to generate a stable reduction in the correlation. To explore this question, we randomly selected three combinations of time (τ_T) and slide windows (τ_s). As described in Figures 8C–E, the correlation presents a trend of increasing followed by decreasing. The variation in correlation arises from the changing rules of the firing rate with the SFA. The output correlation shows a temporary increase for a very short stimulation time, as shown in the upper-right corner of each subgraph in Figures 8C–E. This is due to the rapid firing rate, resulting in shared spike trains of pairwise neurons having more similar parts in the initial stage. However, this immediate increase in the correlation disappeared and is reversed to a decreased result when the stimulation time increased. In the second stage, the SFA decreased the correlation by slowing down the firing rate of individual neurons. The slow firing patterns in the second stage may have an important impact on the population coding of sustained input signals to the neurons.

The output correlation is measured by Pearson's coefficient, resulting in the output correlation being related to τ_T and τ_s (Barreiro et al., 2012). Because only a fraction of spike trains is included in τ_T, the instantaneous partial firing rate may generate significant variability in the output correlation (de la Rocha et al., 2007; Barreiro et al., 2012). Different combinations of τ_s and τ_T are selected to verify whether they affected the decreased results of SFA on the correlation in this study. The chosen time window in this section is 10 times larger than the slide window (Barreiro et al., 2012). As shown in Figure 9A, τ_s is fixed at 100 ms to observe the effect of τ_T on output correlation. The output correlation exhibits a weak increase across the range of values of τ_T. Similar to the variation tendency in Figure 5A, the correlation variations caused by τ_s are not noticeable when the stimulation time is fixed at 10,000 ms in Figure 9B. All results show that τ_s and τ_T have an impact on the numerical value of the correlation. However, their different combinations make no difference to the effect of SFA on reducing the correlation. Stimulation time is the most critical factor affecting the correlation difference between the two graphs in Figure 8. High similarity in the initial part of the spike trains with the SFA significantly affects the correlation for a short stimulation time. The correlated input and the input correlation become the main factor decreasing the correlation over a long stimulation time.

M and AHP currents have been proven to reduce the output correlation by decreasing the firing rate in this study. The Prescott model is a single-compartment conductance-based physical model that neglects other adaptations of the biological model, such as the Na⁺ inactivation K⁺ current. In this section, we explore the effect of the slow inhibitory current (I_KNa) on this correlation. In addition to the adaptation current, the dynamic threshold can generate adaptation and decrease the firing rate.

First, we introduce a typical I_KNa from the Wang model to explore its effect on output correlation (Wang et al., 2003). The red line in Figure 10A indicates an increase in the output correlation, completely different from the decreasing phenomenon in Section 3.1. The sequences generate rapid firing for a long time, even more than 3,000 ms, which makes the front of the spike trains highly similar. The similarity is hardly affected by the random spiking activity in the correlated input. Therefore, the correlation remains very high, close to 1. The firing rate decreased, resulting in the correlation demonstrating a steady decline as the adaptation conductance increased, as shown in Figure 10B. This reduced effect of the adaptation current on the correlation is consistent with the above results for M and AHP currents. We further studied the variation in ISIs. The ISIs exhibited a regular hill without an SFA. As the adaptation conductance increased, the ISIs hill moved toward larger values, as depicted in Figure 10E. This hillside is a significant factor that decreased the correlation. We then calculated the mean value of the ISIs that exhibited a gentle ascent, as shown in Figure 10C. Because the fluctuations in the correlated input dominated the correlation variation, the CV of the ISIs demonstrated a decreasing tendency, as shown in Figure 10D. In conclusion, I_KNa decreased the correlation by decreasing the firing rate.

The M and AHP currents have been proven to decrease the correlation in the Prescott model. However, the same mechanisms may behave differently in different models. We chose the Ermentrout model to verify whether the same SFA mechanisms in the various models had diverse effects on the correlation. Two aspects confirm that the results are independent of the models: the density of the input current and the adaptation conductance. As shown in Figure 11A, the correlation decreases as the adaptation conductance increases. At the same conductance, increasing the input intensity enhanced the output correlation. SFA mechanisms are the primary factors that change the correlation by decreasing the firing rate, as displayed in Figure 11B. Different SFA mechanisms in various models generated similar reduced effects on the correlation. The most remarkable difference between the Ermentrout and the Prescott models is that the correlation (Figure 11C) and firing rate (Figure 11D) in the Ermentrout model reduces smoothly as compared to the correlation change seen with the Prescott model. The downward trend of the output correlation curves is smoother without inflections. The correlation demonstrates continuous variation as g_AHP increases. The variation in firing patterns with AHP current is a continuous change with adaptation conductance changes. Thus, the variation in correlation presents a smooth decrease regardless of whether in the Prescott or Ermentrout model. The decreased correlation is caused by a reduction in firing rate, accompanied by ISIs variations. The ISIs are plotted in Figure 12, which shows that the distribution expands to larger values. In contrast to the variation in the Prescott model, the Ermentrout model is more regular because the firing activity does not stop during the response.

The mechanisms related to SFA introduced above are based on the current mechanism in a physiological phenomenon. Different ionic currents generate various adaptation phenomena. The models all verified that the effect of SFA is to reduce output correlation. This brings to question whether this SFA effect exists in a mathematical, physical model where dynamic thresholds generate SFA. First, we use LIFAC to generate adaptation current and verify whether adaptation current has the effect of decreasing correlation. Then, we test this decreased effect with another model using a dynamic threshold. As shown in Figure 13B ③, the pairwise neurons present a relatively uniform and regular firing pattern without adaptation. The addition of adaptation current makes the firing patterns slow down (Figure 13B ① and ② ). As shown in Figure 13C, the firing rate becomes slower as the threshold increases. We speculate that the adaptation current decreases the correlation by reducing the firing rate. When the dynamic thresholds are applied to the model, the membrane potential fluctuations become sparse, where the membrane potential shows dense distribution at the beginning and then slows down. With the increase in dynamic thresholds, the correlation presents a downward trend; see Figure 13A. Meanwhile, the firing rate (Figure 13D) shows a similar decrease tendency to the correlation in Figure 13C. This variation reflects that the dynamic threshold changes the correlation by decreasing the firing rate. At the same stimulus intensity, a larger adaptation strength generates a smaller correlation resulting from the decreased firing rate and sparse firing patterns. Furthermore, increased input current intensity enlarges the output correlation under each dynamic threshold but is uninfluential to the decreasing correlation tendency. The pairwise neurons with adaptation current or dynamic threshold decrease the correlation when SFA is introduced. The firing rate presents a decrease in the same trend as shown in Figures 14C, D. In addition, the ISI distributions in Figures 14A, B prove that the increase of ISIs accompanies the decrease in correlation. The mean values of ISIs show a smooth ascent in Figures 14C, E, indicating an obvious decrease in the firing rates. On the contrary, the CV of ISIs shows a reduction in Figures 14D, F due to the correlated noise components gradually playing a more critical role in generating firing patterns. These results are in accordance with the previous results in Section 3.2.

In Sections 3.1–3.4, the biological models and the phenomenological models are utilized to investigate the role of SFA in output correlation. The results show that SFA decreases the firing rate and further attenuates the output correlation of pairwise neurons. To make a mechanism analysis, a simple phenomenological model using a non-linear input function f, similar to that used in de la Rocha et al. (2007), is introduced. As shown in Figure 15A, λ₁ and λ₂ are in analogy to the pairwise neurons model. The input correlation is c, which is set to 0.5. The function f is set to threshold-linear. As shown in Figure 15B, the threshold-linear characteristic makes the model a rectifier. SFA decreases the slope of the linear portion of the rectifier. The output correlation is lower with adaptation compared with the output correlation without correlation (Figure 15C). The input distribution P(λ₁, λ₂) and the output distribution P(y₁, y₂) are depicted in Figures 15D, E. Furthermore, the output distribution with adaptation is plotted in Figure 15F. The introduction of adaptation decreases the correlation and response intensity. In addition, another Gaussian input, μ = 0.75 is investigated. The input distribution P(λ₁, λ₂) and the output distribution P(y₁, y₂) with or without adaptation are drawn in Figures 15G–I. The distribution is shifted to the lower parameter area. The eccentricity is smaller than that at μ = 1.5, which indicates that the external inputs play a pivotal role in the output correlation. The results show that when the Gaussian input overlaps with the non-linear area of function f, the output correlation is lower than the input correlation. As the eccentricity becomes lower for decreased input intensity, the output correlation attenuates. The eccentricity depends on the input correlation and the input intensity. In conclusion, SFA, through changing the input non-linearity of single neurons, may play an important role in the degree of correlation in a neural network.