How noise contributes to time-scale invariance of interval timing

Abstract

Time perception in the suprasecond range is crucial for fundamental cognitive processes such as decision making, rate calculation, and planning. In the vast majority of species, behavioral manipulations, and neurophysiological manipulations, interval timing is scale invariant: the time-estimation errors are proportional to the estimated duration. The origin and mechanisms of this fundamental property are unknown. We discuss the computational properties of a circuit consisting of a large number of (input) neural oscillators projecting on a small number of (output) coincidence detector neurons, which allows time to be coded by the pattern of coincidental activation of its inputs. We show that time-scale invariance emerges from the neural noise, such as small fluctuations in the firing patterns of its input neurons and in the errors with which information is encoded and retrieved by its output neurons. In this architecture, time-scale invariance is resistant to manipulations as it depends neither on the details of the input population nor on the distribution probability of noise.

FIG. 1.. (Color online) Accurate and time-scale invariant interval timing.

(A) The response rate of rats timing a 30s interval (left) or 90s interval (right) overlap (center) when normalized by the maximum response rate (vertical axis), respectively, the criterion time (horizontal axis); redrawn from [8]. (B) Time-scale invariance in human subjects for 8s and 21s criteria; redrawn from [10]. (C) Systemic cocaine (COC) administration speeds-up timing proportional (scalar) to the original criteria 30s and 90s; redrawn from [8]. (D) The hemodynamic response associated with a subject’s active time reproduction scales with the timed criterion, 11s v. 17s; redrawn from [14].

FIG. 2.. (Color online) The neurobiological structures involved in interval timing and the corresponding simplified SBF architecture.

(A) Schematic representation of some neurobiological structures involved in interval timing. The color-coded connectivities among different areas emphasize appropriate neuromodulatory pathways. The two main areas involved in interval timing are frontal cortex (FC) and basal ganglia (BG). (B) In our implementation of the SBF model, the states of the N cortical oscillators (input neurons) at reinforcement time T are stored in the reference memory as a set of weights w_i(T). During test trials, the working memory stores the current weights w_i(t) and, together with the reference memory, projects its content onto spiny (output) neurons of the BG. FC: frontal cortex; MC: motor cortex; BG: basal ganglia; TH: thalamus. GPE: globus pallidus external; GPI: globus pallidus internal; STn: subthalamic nucleus; SNc/r: substantia nigra pars compacta/reticulata; VTA: ventral tegmental area; Glu: glutamate; DA: dopamine; GABA: gamma-aminobutyric acid; ACh: acetylcholine.

FIG. 3.. (Color online) A noise-free SBF model does not produce time-scale invariance.

(A) Analytically predicted (inset) and numerically generated output of a noise-free SBF-sin model with N = 100 for T = 5s and T = 15s. (B) As predicted analytically (theoretical), the width the output function of a noise free SBF-sin model is independent of the criterion time. (C) A noise-free SBF-ML model does not produce time-scale invariance either. The width of the Gaussian envelopes (dashed line for T = 5s and dashed-dotted line for t = 15s) remains constant.

FIG. 4.. (Color online) Time-scale invariance emerges from criterion time noise in the SBF model.

(A) Time-scale invariance emerges spontaneously in a nosy SBF-sin model; here the two criteria are T = 30s and T = 90s. The output functions (continuous lines) were fitted with Gaussian curves (dashed lines) in order to estimate the position of the peak and the width of the output function. (B) In an SBF-sin model, the standard deviation increases linearly with the criterion time in all four trials shown with different symbols. (C) In an SBF-ML implementation, low levels of Gaussian noise (solid rectangles represent σ_T = .1% and circles σ_T = 1%) produce an almost constant standard deviation of the output function similar to noise free case. At high enough levels of noise (solid triangles represent σ_T = 10%), the SBF-ML model with criterion time variance produces a standard deviation σ_o that increases linearly with the criterion time T, which is the hallmark of time-scale invariance.

FIG. 5.. (Color online) Time-scale invariance is robust to noise manipulation in the SBF model.

(A) In the presence of both criterion time and frequency variance, the SBF-ML produces accurate and time-scale invariant output. At the same time, multiple sources of noise determine a long tail in the output function, which is similar with experimental findings. (B) The scalar property is also preserved regardless the type of criterion time variability (uniform noise - solid triangles, Poisson noise - solid squares).