Time-scale invariance as an emergent property in a perceptron with realistic, noisy neurons

Abstract

In most species, interval timing is time-scale invariant: errors in time estimation scale up linearly with the estimated duration. In mammals, time-scale invariance is ubiquitous over behavioral, lesion, and pharmacological manipulations. For example, dopaminergic drugs induce an immediate, whereas cholinergic drugs induce a gradual, scalar change in timing. Behavioral theories posit that time-scale invariance derives from particular computations, rules, or coding schemes. In contrast, we discuss a simple neural circuit, the perceptron, whose output neurons fire in a clockwise fashion based on the pattern of coincidental activation of its input neurons. We show numerically that time-scale invariance emerges spontaneously in a perceptron with realistic neurons, in the presence of noise. Under the assumption that dopaminergic drugs modulate the firing of input neurons, and that cholinergic drugs modulate the memory representation of the criterion time, we show that a perceptron with realistic neurons reproduces the pharmacological clock and memory patterns, and their time-scale invariance, in the presence of noise. These results suggest that rather than being a signature of higher order cognitive processes or specific computations related to timing, time-scale invariance may spontaneously emerge in a massively connected brain from the intrinsic noise of neurons and circuits, thus providing the simplest explanation for the ubiquity of scale invariance of interval timing.

Figure 1. The ubiquity of scale-invariant timing in humans

(A) In human adults, interval timing functions peak at the target duration, and their width increases with duration (left), such that they superimpose in relative time units, indicating scale-invariant timing (right) (adapted from Wearden, et al., 1997). (B) Scale-invariant timing is ubiquitous throughout human development, in 3-year old (left), 5-year old (center), and 8-year old children (right) (adapted from Droit-Volet, et al., 2001). (C) Scale-invariance of timing does not depend on the modality of the timed stimulus (adapted from Zarco, et al., 2009). (D) Scale-invariant timing is ubiquitous in both explicit (generalization) and implicit (estimation) timing tasks (adapted from Piras & Coull, 2011). (E) Scale-invariant timing is ubiquitous in timing tasks with different motor requirements, and number of to-be-timed signals: production of single (STT) or multiple time intervals (MTT), temporal categorization (CAT) and discrimination (DIS) (adapted from Merchant, et al., 2008).

Figure 2. Time-scale invariance in primates and rodents

(A) Response rate in peak-interval experiments with human adults trained with three target durations (8s, 12s, 21s) (Rakitin, et al., 1998). (B) Responses from (A) rescaled both relative to the target duration and the maximum response rate. (C) Percent maximum response rate in peak-interval experiments with rats trained with two target durations (30s, 90s); administration of indirect dopamine agonist cocaine (COC) results in an immediate, scalar (proportional) leftward shift in response functions, relative to saline (SAL) (adapted from Matell, et al., 2004). (D) Scalar timing in Rhesus monkeys (adapted from Zarco, et al., 2009).

Figure 3. The perceptron

(A) Schematic representation of the neurobiological structures involved in interval timing in the Striatal Beat Frequency model (SBF) model (Matell & Meck, 2004). Frontal oscillators are implemented as biophysically realistic ML neurons. ACh: acetylcholine; FC: frontal cortex; BG: basal ganglia; DA: dopamine; Glu: glutamate; GP_E: globus pallidus external; GP_I: globus pallidus internal; STn: subthalamic nucleus; SNc/r: substantia nigra pars compacta/reticulata; TH: thalamus; VTA: ventral tegmental area. (B) The basic structure of the perceptron used to implement the SBF-ML model. Dashed lines in panel (A) signify couplings that were not implemented in our SBF-ML version (Oprisan & Buhusi, 2011).

Figure 4. An SBF model with phase oscillators and no variance does not exhibit time-scale invariance

(A) Normalized output function in numerical simulations of the SBF model versus theoretical predictions (inset). (B) The standard deviation of the Gaussian envelope of the normalized output function is independent of the criterion time in the absence of any variability in model parameters, indicating lack of time-scale invariance.

Figure 5. Time scale invariance emerges in an SBF model with phase oscillators upon introducing noise

(A) Normalized output functions (continuous line) in numerical simulations of the SBF model with phase oscillators affected by uniform noise for three different criteria show a significant increase in the standard deviation with the criterion time (T₁ = 15s, T₂ = 30s, and T₃ = 45s). The corresponding Gaussian envelopes are shown with dashed lines. (B) Standard deviation varies linearly with criterion time, indicating the noisy SBF model exhibits scale-invariant timing, both for Gaussian (solid squares) and uniform (solid triangles) distribution of noise.

Figure 6. Time-scale invariance in an SBF-ML model with noisy Morris-Lecar neurons

(A–C) For very low memory variance, the standard deviation of the Gaussian fit (shaded area in panels AC) varies very little with criterion time failing to indicate scale-invariant timing. (D) The scalar property emerges only when the system is noisy (at large variances). All simulations used 600 cortical oscillators in the frequency range from 5.5 Hz to 11.5 Hz.

Figure 7. The clock pattern is time-scale invariant

The clock pattern of dopaminergic (DA) drugs (adapted from Meck, 1996): Two groups of rats were trained off-drug to time a criterion time of either 40s (upper pattern) or 20s (lower pattern); they were then administered either DA agonists or antagonists for 7 sessions, followed by 7 session off drug. The first administration of DA drugs results in an immediate, dose-dependent shift in timing, leftward (faster timing) for DA agonists (solid squares, methamphetamine), and rightward (slower timing) for DA antagonists (solid circles, haloperidol). Under continuous training with the pre-drug criterion time and despite continuing the drug administration, the timing functions recalibrate to the pre-drug criterion time. Upon discontinuing the drug, timing functions immediately rebound in the opposite direction, then gradually recalibrate to the pre-drug criterion time (Meck, 1996). Solid triangles indicate numerical simulations with the SBF-ML model (Oprisan & Buhusi, 2011). Insets: The insets indicate the response function generated by the SBF-ML model throughout the clock pattern (indicated by arrows, and by a triangle symbol of the color of the inset) (Oprisan & Buhusi, 2011). (A1) Immediate rebound from T = 40 s to T^** = 48 s upon discontinuing methamphetamine. (A2) Recalibration under methamphetamine. (A3) Immediate shift under methamphetamine from T = 40 s to T^* = 32 s. (A4) Immediate shift under haloperidol from T = 20 s to T^* = 24 s. (A5) Recalibration under haloperidol. (A6) Immediate rebound upon discontinuing haloperidol. The dashed (panels A1–A3), respectively, continuous (panels A4–A6) smooth lines represent Gaussian fits. Note that the effect of the same dose of drug is twice as large in the 40s rats (upper pattern) than in the 20s rats (lower pattern); also, in all insets, the width of the function is proportional to the peak time, indicative of time-scale invariance in both the behavioral and pharmacological dimensions of the experiment. Reproduced from (Oprisan & Buhusi, 2011).

Figure 8. The memory pattern is time-scale invariant

The memory pattern of cholinergic (ACh) drugs (adapted from Meck, 1996): Two groups of rats were trained off-drug to time a criterion time of either 40s (upper pattern) or 20s (lower pattern); they were then administered either ACh agonists or antagonists for 7 sessions, followed by 7 session off-drug. The first administration of ACh drugs results in a minimal effect; repeated ACh drug administration results in a gradual, dose-dependent shift in timing, leftward for ACh agonists (solid circles, physostigmine), and rightward for ACh antagonists (solid squares, atropine). Upon discontinuing the drug, timing functions gradually recalibrate to the initial criterion time (Meck, 1996). Solid triangles indicate numerical simulations obtained with the SBF-ML model (Oprisan & Buhusi, 2011). Insets: The insets indicate the output function generated by the SBF-ML model with biophysically realistic ML neurons throughout the memory pattern (indicated by arrows, and by a triangle symbol of the color of the inset) (Oprisan & Buhusi, 2011). (A1) Gradual shift from T = 40 s to T^* = 50 s under atropine. (A2) Gradual recalibration upon discontinuing atropine. (A3) Gradual recalibration upon discontinuing physostigmine. (A4) Gradual recalibration under physostigmine. The dashed (panels A1–A2), respectively, continuous (panels A3–A4) smooth lines represent Gaussian fits. Note that the effect of the same dose of drug is twice as large in the 40s rats (upper pattern) than in the 20s rats (lower pattern); also, in all insets, the width of the function is proportional to the peak time, indicative of time-scale invariance in both the behavioral and pharmacological dimensions of the experiment. Reproduced from (Oprisan & Buhusi, 2011).