Application of instrumental investigations of cosmic-ray muon arrival time measurements to human brain processing of subjective temporal order

doi:10.21203/rs.3.rs-2456854/v1

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Application of instrumental investigations of cosmic-ray muon arrival time measurements to human brain processing of subjective temporal order

https://doi.org/10.21203/rs.3.rs-2456854/v1

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Since the 18th century, it has been widely accepted that causal order is independent from notions of time. Recent neurological experiments have shown that it is possible to violate subjective temporal order between motor-driven events and sensory events. This violation, subjective temporal order reversal, has been explained by the recalibration of the timeline to keep causality. However, subjective temporal order could also be violated between non-causal events. Therefore, it might be more appropriate to base a new approach upon another trivial condition. In this work, a condition called the consistency of event sharing is employed. An event shared by 2 individuals (A and B) at the same location must be consistent between these individuals (e.g., if A shakes B's hand, the moment when A touches B's hand is exactly the same moment when B touches A's hand). First, the timeline recalibration was defined and formulated in an instrument-muon system under this condition. Then, the consistency of event sharing rule was applied to prior neurological experiments. As a result, it was found that this condition sufficiently explains the reversed subjective causal order.

Recognition of temporal and causal order is an essential component of the episodic memory that controls “mental time travel”¹. Temporal order and simultaneity have been extensively discussed in various fields of physics^2,3,4,5, philosophy⁶, and even in science fiction⁷. In particular, temporal order judgement (TOJ) ability and simultaneity judgement ability are extensively studied in the field of neuroscience with a focus on the process of aging^8,9,10,11,12 and neurological disorders^{13,14,15,16,17}. Experimental studies of "time" and "timing" can provide information about the neuronal basis of human cognitive functions^18,19. The general point of view regarding temporal order and simultaneity can be summarized with the following example: if A meets B first and subsequently A meets B again, the temporal order of these two simultaneous events (SEs) must be the same for A and B in order to correspond to everyday logic; namely, B should also meet A first and subsequently, B should also meet A again. However, if the sequence of these SEs is different between A and B, A will tend to believe that B’s temporal order is reversed and vice versa. Subjective temporal order could be reversed due to the effects of multi-sensory influences²⁰, aging^{21,22,23,24,25}, and neurological impairments^26,27.

Temporal information processing (TIP) is an essential feature of many aspects of human cognition for time-sequential events, and many studies have revealed TIP deficiencies in members of the elderly population²⁸. Thus far, a number of experimental data have indicated that subjective time varies depending on our TIP ability. Evidence for this has come from experiments having to do with age-related changes in the perception of temporal order²⁹, gap detection^30,31,32,33, duration discrimination^34,35,36,37, temporal bisection^38,39, and time estimation^40,41,42,43. For example, Philips et al.⁴⁴ found that in general, elderly people (aged approx. 72 years) needed longer interstimulus intervals (ISIs) than younger people (aged approx. 22 years) to correctly report the temporal order of two diode illumination events. A similar conclusion was obtained from experiments on auditory perception^45,46. However, the mechanism responsible for deterioration of temporal order processing has not yet been totally clarified.

Recently, it was found that subjective temporal order can also be reversed; by repeatedly injecting a constant delay between a motor-driven event and a sensory event, the subjective temporal order of the action and the resultant event cognition could be reversed when the resultant event is suddenly presented with an unexpected shorter delay⁴⁷. It had previously been interpreted that this reversed order was illusional, and was merely created by these consistent lag actions which were responsible for triggering a recalibration of the human brain’s timeline in order to conserve causality initiated by these lag actions. Currently, this interpretation can be more comprehensively explained by the notion that the human brain subconsciously segregates and ignores any information that violates causal order^48,49. However, subjective temporal order could also be violated between cross-sensory events⁴⁷. This experimental fact indicates that causal order conservation cannot be universally applied to explain this kind of temporal order reversal. It might be more appropriate to base a new approach upon a trivial and more universally applicable binding condition than causal order conservation.

In order to objectify human's perception and cognition of temporal and causal order, psychologists have developed extensive methodologies and statistical tests. In this paper, an instrumental particle physics approach is proposed as a complimentary approach to these existing strategies. Since physical experiments adapt more easily to a theory-ladenness approach, a formulation based upon physical experiments could be used for inferring underlying physical rules or principles concretely. Although physical experimental results cannot be postulated to directly match all the complex facets of humans’ perception and cognition, the value of finding similarities between physical experiments and psychological experiments is to provide a different perspective than would be possible with other conventional strategies. In this paper, first, the time-sequential relativistic events are investigated to formulate the instrumental timeline recalibration under a condition: the consistency of event sharing that is also applicable to non-causal time-sequential events. Subjective temporal order reversal observed in Stetson's experiment will be explained with this new approach.

In this section, we show that the timeline recalibration is a simple logical deduction to keep the consistency of event sharing between the instrument and particles which have different rate of time progress. The methodology discussed here will be used in the next section to infer a psychological case. Our instrumental approach was modeled by timing measurements of naturally occurring cosmic-ray muon events detected with 2 vertically aligned detectors. Cosmic-ray muons are in general strongly relativistic particles, traveling at almost the speed of light in a vacuum (c) through any kind of material; moreover, their speed is not affected by the media condition they travel through as long as the energy of the muon is within the relativistic range. Muons are 205 times heavier than electrons thus, radiative interactions with matter are suppressed, and they can penetrate into subsurface regions of the Earth. By utilizing the relativistic and penetrative nature of cosmic-ray muons, imaging of gigantic bodies such as volcanoes⁵⁰, pyramids⁵¹, tsunamis⁵², cyclones⁵³ and other underground structures as well as underwater/underground positioning⁵⁴ and navigation⁵⁵, precise time synchronization⁵⁶, and secured communication⁵⁷ have been made possible. As can be seen in Fig. 1, most cosmic-ray muons have a Lorentz factor (γ) much larger than 1, and we can approximate that cosmic ray muons travel at ~ c⁵⁴. Accordingly, the decay constant (2.2 µs) of these muons is extended by a factor of γ.

Cosmic-ray muons are produced in the air via the collision between primary cosmic rays and nuclei in the atmosphere, and arrive only from the upper hemisphere. Therefore, with the aforementioned detector configuration, the muons always pass through the top detector first and the bottom detector second. Reference clock pulses are utilized to open and close gates. Figure 2A shows the time series of logic pulses that are used as “gates” for muon detection; when the gate is opened, having a status of Logic level − 1, it is ready to receive muon signals. Logic level 0 indicates that the gate is closed and any signals are discarded. The pulse width and the pulse period are respectively τ and 2τ, where τ is the muon's time of flight between 2 detectors from the observer's point of view. The distance between the detectors is arbitrary; hence τ is also arbitrary. The bottom detector is gated by the gate and delay generator (GDG). When the top detector's gate is opened, GDG provides a delay of τ, and furnishes an output logic pulse with negative polarity (Logic level − 1) with a width of τ. The GDG concept will also be used in the discussion section. In this measurement, the same gating condition is repeated. In other words, the same gating condition is retrospected/prospected in the past/future. With this gating condition, 50% of the total muons that passes through both the top and the bottom detector will be detected in the instrument's frame. The consistency of event sharing requires that this fraction must be the same also in the muon's frame. However, if timeline recalibration is not applied, since (A) the clock pulse width is extended by a factor of γ, and (B) due to the Lorentz contraction, the distance between these detectors would be contracted by a factor of γ⁻¹ and consequently the bottom detector's gate would be closed for many of the muons (Fig. 2B). This inconsistency can be explained in the following way: the top detector and the bottom detector are synchronized in the instrument's frame, but they are not synchronized in the muon's frame (See Method Section). However, in this work, we do not simply correct the frame-dependent distant synchronicity in an a priori way, but we try to find a more generalized formulation to hold only the consistency of event sharing for the purpose of a neurological application described later.

A gate opening event shared between the instrument and the muon at the same location (the bottom detector) must be consistent between the instrument and the muon. Consequently, the bottom detector's timeline has to be shifted by a factor of:

|δT| = τ (γ ± γ⁻¹) + nτ, (1–1)

where δT is the recalibration offset (RO), and n is an integer that holds n ≥ 0. In Eq. (1–1), there are two options: τ(γ - γ⁻¹) + nτ and τ(γ + γ⁻¹) + nτ. The sign (plus or minus) depends on the recalibration direction (either forward or backward) of the muon's time series. τ(γ - γ⁻¹) + nτ would shift the timeline at bottom detector towards the past (a backward offset). τ (γ +γ⁻¹) + nτ would shift the timeline at the bottom detector towards the future (a forward offset). If we express the shifting direction of the RO by the positive (forward) and negative (backward) signs assigned to δT, Eq. (1–1) will be:

±δT = ±τ (γ ±γ⁻¹) ± nτ, (1–2)

The next argument can focus on the "±" degree of freedom (DOF). For cosmic muons, the direction of temporal order is always defined as starting from the top detector and ending at the bottom detector. The forward offset changes this order. More specifically, if the gates are numbered sequentially from the start of the measurement, the muon passes the Nth gate at the top detector and the (N-n)th gate (the past gate) at the bottom detector in the muon's frame; in the instrument's frame, the muon passes the Nth gate both at the top and the bottom detectors. Therefore, if we apply temporal order conservation, the option + δT = τ(γ +γ⁻¹) + nτ would be discarded. However, "n" DOF still remains. There is another way to restrict "±" DOF. If we assume that the path of least resistance: the principle written by Pierre Louis Maupertuis: "Nature is thrifty in all its actions" (or, in other words, the idea that nature tends to choose the “shortest/least resistance” (spatiotemporal) path of action) is applicable to choose a more reasonable and shorter offset, then option τ (γ - γ⁻¹) is uniquely chosen (See Method Section). For the application to TOJ testing which will be described later, it is more convenient to apply a condition independent from temporal order.

The RO can be expressed in the (passage of time) - (time interval) space (Fig. 3). When studying muons' kinematics with a theoretical approach, event occurrence in the inertial frame is conceptually described by a continuous line (worldline) defined in each frame. Likewise, synchrony is also defined in each frame; hence tilting the worldline (the dashed blue line in Fig. 3) can compensate for non-synchrony in another frame at a distance. On the other hand, with the instrumental approach, this frame-dependent synchrony is compensated by recalibrating the timeline by a factor of τ (γ - γ⁻¹), which can be achieved by tilting the time zero line (TZL) instead of tilting the muon's worldline. TZL is defined by the measurement start time (the dashed green line in Fig. 3). Although in physics, both approaches bring the same result, the latter concept can be more naturally postulated to a specific psychological case discussed in the current work. But in a psychological case, the passage of time and time interval are treated independently^59,60. Therefore, it is more convenient for us to generalize Eq. (1–1) to:

|δT| = τ (γ₁−γ₂), (1–3)

with 2 independent parameters: γ₁and γ₂. In the next section, by respectively replacing the instrument and the muon with a human being individual and a signal, this formulation is applied to evaluating differences in the subjective duration of events in a human's frsme and the signal's frame.

Now we aim to discuss an example of a possible application of this formulation to explain prior psychological experimental results. Psychological time has been an important topic of research for more than a century^61,62. Early theorists speculated about possible differences in duration judgments between younger and older adults. James (1890)⁶³ claimed that ''the same space of time seems shorter as we grow older." Mach (1900) described that the psychological unit of time lengthened with age. There are several kinds of models to explain duration judgments⁶¹. One model is based on an internal clock that may be influenced by biological variables such as metabolism and brain temperature, etc.⁶⁴. The slowing of a biological clock would presumably give rise to the feeling that external time passes faster. Another kind of model proposes that duration judgements are cognitive constructions that are influenced by attention and memory processes⁶¹. Several kinds of methodological and individual factors may influence duration judgments due to their effects on attention and memory^61,65,66. The ratio of the subjective duration of time (how long an individual perceives an hour to be) to the objective duration of time (the actual duration of an hour) is called the duration judgement ratio (DJR). This is a standard measure calculated and reported in many “duration judgement” studies⁶⁷. An individual with lower DJR experiences time as being contracted and an individual with higher DJR experiences time as being expanded. Different circumstances can change DJR. For example, in the same surroundings one person may have high DJR when paying close attention to time, but another person may have low DJR in a state of distraction and time would seem to contract to them. Intentional binding is another factor that influences DJR. As mentioned previously, humans also tend to have a general impression that time passes more quickly as we grow older. Friedman and Janssen tried to quantify this impression by constructing a long-term (hour to year) memory model⁶⁸, but it seems that this problem is too complicated for this model to comprehensively and satisfactory uncover the mechanism controlling the variations in DJR.

On the contrary, little by little, subjective temporal order in shorter timescales (as experienced by humans) has been getting more quantified, and can be systematized in a physical way. Stetson et al.⁴⁷ suggested that dynamic temporal order recalibration is needed to define temporal order of actions and sensations. Their experiment involved 2 participants: the individual and the instrument. (Hereafter individual defined in this experiment is written in italic text in order to distinguish this label from an individual as a generic term.) The individual is a human and the instrument is a machine. The individual presses a key (motor-driven event) to activate a flash event (sensory event). This recurring flash event would at first consistently appear with a fixed delay (τ₀) (TEST I). After several fixed-delay-flash event cycles, the flash would be suddenly presented with a shorter delay (τ) (TEST II) and this event would tend to be perceived as appearing before the button press (a subjective reversal of temporal order). The point of subjective simultaneity (PSS) is an indicator created by this group to evaluate a particular human brain’s subjective timing error⁴⁷. A PSS equal to 0 would indicate that motor and sensory events would be perceived as synchronous when they physically occur at the same time. In their motor-sensory test, PSS was positive, indicating that it would be possible for an event which occurred in the past and an event occurring in the present according to the objective instrument could be perceived as subjectively simultaneous by the individual. Stetson et al. concluded that "temporal order judgements between motor acts and sensory events dynamically change in order to keep causality assessments appropriately calibrated." However, subjective temporal order reversal was observed also between cross-sensory events⁴⁷ (such as automatic tapping of the individual's finger by the key instead of a keypress initiated by the individual). Since the individual did not cause the following event, there is no causal relationship between these events. Therefore, it seems unreasonable to use the causality as a condition to construct this argument. In contrast, employing a premise that relies on the consistency of events shared by the individual and the instrument (consistency of event sharing) seems more trivial and reasonable.

With an analogy of the previous muon-instrument discussion, Stetson's experimental results can be reformulated as follows: as the individual becomes accustomed to a certain duration of fixed delay by experiencing recurrent motor-sensory (or cross-sensory) feedback, the amount of the delay (τ_expect) of the individual's GDG is fixed by sensory gating with their own internal clock to be ~ τ₀. Sensory gating is the brain's natural response to attenuate irrelevant stimuli^69,70. Accordingly, Stetson's experiment can be rephrased in the following way. The individual presses the key which triggers an opening the instrumental gate to allow the instrument to accept the individual signal, and the instrument detects this individual signal. In this case, there is a causal relationship between the keypress and the instrumental gate opening. Next, the instrument opens the individual gate so that the individual can receive the instrumental signal. Figure 4 shows the time series of the gate opening/closing corresponding to the time series shown in Fig. 2. With this experimental configuration, the individual and instrument continuously emit signals respectively to the instrument and individual, but they can receive these signals only when their gates are opened.

The instrument opens the individual's gate at τ₀ (in the frame of signal) after the instrument receives the individual signal. This process is repeated several times. Then, abruptly, the delay τ₀ changes to τ (in the frame of signal) which is much shorter in duration. SE1 for the individual is defined as these 2 events: the individual's opening the instrumental gate (keypress event) (A1) and the individual signal departure to the instrument (B1). SE2 for the individual is defined as: the instrumental signal arrival at the individual (A2), and the individual's cognition of the instrumental signal (B2). The time required for the signals to travel from the individual to the instrument is negligible. The condition: consistency of event sharing requires that the following relationship:

t(A1)- t(B1) = 0, (2 − 1)

t(A2)- t(B2) = 0, (2–2)

holds both in the individual's frame and the signal's frame, where t(A1), t(B1), t(A2), and t(B2) are respectively the moments when A1, B1, A2, and B2 occur, namely:

[t(A1)- t(B1)]_individual - [t(A1)- t(B1)]_signal =0, (2–3)

[t(A2)- t(B2)]_individual - [t(A1)- t(B1)]_signal =0. (2–4)

Eqs. (2–3) and (2–4) also must hold for non-causal events.

Next, the individual's internal GDG is modeled according to the following description: (A) “Fixed delay (τ₀)” is the objective time interval (labeled as “instrument” as shown in Fig. 4A) between SE1 and SE2. On the other hand, the corresponding subjective time interval (labeled as “individual”) is defined as the interval between SE1 and SE2; this subjective motor-sensory time interval is determined by the time interval (κτ₀), where κ is a compression factor that is a function of τ₀⁷¹. Previous works have quantified that our perception of when an event occurs depends on the influence of intentional binding (IB) which is the temporal compression that occurs when the brain makes a connection between a voluntary action and its ensuing sensory effect⁷². However, in order to make the current modeling simpler, the IB effect is not considered, i.e., κ ~ 1. The individual is recurrently exposed to this delayed sensory feedback, and eventually, the amount of delay of the individual's internal GDG is set to be τ₀ (Fig. 4B red dashed lines). (B) “Variable delay (τ)” (Fig. 4A) is the objective interval between SE1 and SE2 after exposure to recurrent fixed delayed sensory feedback.

The recalibration offset defined in the previous section is re-labeled as recalibration memory offset (RMO) since in the case of the individual, the past events are all stored in their memory, and are expressed with only 2 parameters: γ₁ and γ₂. The moment when the individual observed the signal is defined as time zero (TZ) (Fig. 4), and here Eq. (2–3) must hold. However, when τ≠ τ₀, Eq. (2–3) doesn't hold without the RMO (Fig. 4B), thus from Eq. (1–3), the memory timeline must be shifted with a magnitude of:

δT = τ₀(γ₁-τ/τ₀), (3)

to hold Eq. (2–3) (Fig. 4C), where the RMO is always associated with the forward direction as long as τ₀-τ > τ. Here for simplicity, variations in the subjective passage of time was not considered. Therefore, γ₁ = 1. Accordingly, the memory of the individual's opening the instrumental gate (key press event) is shifted forward. As a consequence, the temporal order between the memory of the keypress event and the moment when the individual perceived the instrumental signal is reversed. As was suggested in the reference⁴⁷, it is also possible to explain this reversal as resulting from the correction of the causal simultaneity between the keypress event and the instrumental gate opening event. However, it is difficult to conclude that causality is the main driving factor of this recalibration. Since the memory of the keypress event is also shifted forward along with the instrumental gate during this timeline correction process, this subjective causal simultaneity is still violated. Moreover, the non-zero PSS found in Stetson's experiment showed that subjective temporal order is reversed also for cross-sensory events; hence this reversal occurs for non-causal events⁴⁷. Therefore, it is reasonable to explain that this temporal order reversal was the result of that the individual tried to keep the consistency of event sharing: the gate opening event must be shared by the individual and the signal at exactly the same time as long as they are located exactly at the same location. The outcome is that the subjective temporal order reversal is a natural consequence of the binding condition: the consistency of event sharing.

The condition: consistency of event sharing is trivially applicable to both the individual and the instrument (and also to all instruments and human beings). It is also reasonable to assume that the path of least resistance is applicable to the neurological process of human beings. In order to confirm this effect is present, the following experiment is proposed. Figure 5 shows a hypothetical experimental scheme to confirm the path of least resistance. In this configuration, the individual send the individual signals to the instrument twice, and the instrument receives these two temporally distinguishable signals (SE1 and SE2). The time interval between these two actions is shorter than τ₀. When the instrument receives these two individual signals, the instrument opens the individual gates after different delays (τ₁ and τ₂), and the individual receives two identical but temporally distinguishable signals (SE3 and SE4). τ₀,τ₁, and τ₂hold the following relationship: τ₀-τ₂ > τ₂, τ₀-τ₁ > τ₁, and τ₁ > τ₂. Since these two signals are identical, the individual cannot distinguish which signal was triggered by which action. If the path of least resistance is applicable, the RMO is optimized so that the action is minimized. In this specific case, the RMO shift is not needed (Fig. 5B). On the other hand, if the path of least resistance is not applied, other options to keep consistency of event sharing exist and consequently, subjective causal order will be occasionally reversed (Fig. 5C).

In reality, RMO fluctuates due to several reasons: (A) the amount of delay of the internal GDG (τ_expect) may depend on individuals; (B) τ_expect may fluctuate as a function of time; (C) the internal GDG may not always function during the measurement; (D) the parameters (A)-(C) may depend on the individual's group we chose. There are other several possible reasons for the fluctuation of the RMO. Therefore, it is convenient to define the averaged RMO, the RMO averaged over the measurement time or the RMO averaged over a certain individual's group (for example, a certain age group).

Vercillo et al.⁷³ reported that children (< 8 years old) do not recalibrate motor-sensory temporal order after exposure to delayed sensory feedback. This result indicates that the children's averaged RMO is smaller than the adult's averaged RMO. Therefore, based on our formulation, it is inferred that the fraction of individuals with their own GDGs is smaller in the children's group than in the adult's group. Davies et al.⁷⁴ reported that normal adults and typical children show significant differences in sensory gating (testing stimulus)/(conditioning stimulus)(T/C) ratios. A small T/C ratio indicates good sensory gating ability. The report by Davies et al. showed that the typical children's T/C ratio is significantly larger (almost doubled in the N100 component) than the adult's T/C ratio, indicating that the children's GDG doesn't function well; hence, children's averaged RMO is significantly smaller than the adult's averaged RMO.

In conclusion, subjective temporal order was studied for the first time from the instrumental particle physics point of view. Based on the phenomenological investigation with high-energy muon arrival time measurements in different locations, it was found that the timeline defined in a different frame has to be shifted either forward or backward under the condition: consistency of event sharing. We called this shift a recalibration offset. This recalibration offset was defined in the (passage of time)-(time interval) space in conjunction with the path of least resistance. This formulation was applied to the time recalibration of human memory. Consequently, it was derived that reversal of the subjective temporal order is a natural consequence under this condition. Our results indicate different conclusions from the ones in Kant’s Prolegomena which states that causality is a priori; according to this idea, there cannot be particular causal laws that cause specific effects, meaning causal order is absolute, independent from individual perceptions, and cannot be explained by any rule. While this might be true for objective causality which doesn't include individual perceptions, its validity breaks down in some subjective situations. The current work infers that subjective causal law is based on the consistency of event sharing. Consequently, it was concluded that it is more reasonable to explain that the human brain subconsciously segregates any information that violates the consistency of event sharing rather than causality and human memories can be optimized according to the path of least resistance. As a result, the subjective temporal order is occasionally reversed. A valuable next step in this research might be to conduct an experiment to measure the PSS shift for the individuals having different (A) sensory gating T/C ratios and (B) subjective passage of time (γ₁ ≠ 1) perceptions.

δT = τ (γ - γ⁻¹) can be separately derived by modifying Lorentz transformation (Poincaré, 1905)⁷⁵. If we assume that simultaneous events at t₁ = t₂ and at x₁and x₂, from Lorentz transformation:

t'=γ (t + vxc^− 2), (A-1)

x'=γ (x + vt), (A-2)

the same formulation can be derived such that:

δT

=t'₂-t'₁, (A-3)

=γv(x₂-x₁)c⁻², (A-4)

=γv(x₂-x₁)(1-γ⁻²)v^− 2, (A-5)

= τ(γ-γ⁻¹), (A-6)

where β = vc^− 1, then given that x₁ ≠ x₂, there is a timing discrepancy of τ(γ-γ⁻¹) between frames.

Contributions

H.K.M.T. wrote the text. H.K.M.T. prepared the figures. H.K.M.T. reviewed the manuscript.

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Competing interests

The authors declare no competing interests.

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Application of instrumental investigations of cosmic-ray muon arrival time measurements to human brain processing of subjective temporal order

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