Department of Physics and Astronomy, College of Charleston, Charleston, SC, USA

Department of Psychology, Utah State University, Logan, UT, USA

Abstract

Background

The ability to estimate durations in the seconds-to-minutes range - interval timing - is essential for survival, adaptation and its impairment leads to severe cognitive and/or motor dysfunctions. The response rate near a memorized duration has a Gaussian shape centered on the to-be-timed interval (criterion time). The width of the Gaussian-like distribution of responses increases linearly with the criterion time, i.e., interval timing obeys the scalar property.

Results

We presented analytical and numerical results based on the striatal beat frequency (SBF) model showing that parameter variability (noise) mimics behavioral data. A key functional block of the SBF model is the set of oscillators that provide the time base for the entire timing network. The implementation of the oscillators block as simplified phase (cosine) oscillators has the additional advantage that is analytically tractable. We also checked numerically that the scalar property emerges in the presence of memory variability by using biophysically realistic Morris-Lecar oscillators. First, we predicted analytically and tested numerically that in a noise-free SBF model the output function could be approximated by a Gaussian. However, in a noise-free SBF model the width of the Gaussian envelope is independent of the criterion time, which violates the scalar property. We showed analytically and verified numerically that small fluctuations of the memorized criterion time leads to scalar property of interval timing.

Conclusions

Noise is ubiquitous in the form of small fluctuations of intrinsic frequencies of the neural oscillators, the errors in recording/retrieving stored information related to criterion time, fluctuation in neurotransmitters’ concentration, etc. Our model suggests that the biological noise plays an essential functional role in the SBF interval timing.

Background

The capability of perceiving and using the passage of time in the seconds-to-minutes range (interval timing) is essential for survival and adaptation, and its impairment leads to severe cognitive and motor dysfunctions

Interval timing is precise and scalar

**Interval timing is precise and scalar.** Normalized mean lever-press response rate in peak-interval experiments with rats trained with a criterion time of 30s **(A)**, respectively, 90s (**C**; re-drawn from **(B)**.

Although the localization of brain regions involved in interval timing is not yet clear, some progress has been made. For example, both temporal production and temporal perception are strongly connected to striatum and its afferent projections from the substantia nigra pars compacta

Studies done in humans also support the hypothesis that striatum and its afferents are involved in interval timing

The connectionist model is among the first attempts to integrate a large collection of experimental findings into a coherent distributed network model of interval timing by Church and Broadbent

Another successful distributed network model, called the beat frequency model, uses beats between multiple oscillators to produce a much wider range of durations than the intrinsic periods of individual oscillators

In this study, we generalized previous results regarding the quasi-Gaussian shape and the scalar property using the SBF model

We also showed that the scalar property is a universal feature of any SBF model regardless the type of model neurons used and the type of probability distribution functions (

Within the SBF paradigm we used a simple model of cortical oscillators, i.e., a cosine wave (phase) model (see

Methods

We introduced a minimal block diagram that mimics the contributions of some of the neuroanatomical regions known to be involved in interval timing as identified in the Introduction. The schematic diagram includes the following blocks (see Figure **(OSC)**, presumably mimicking the neural oscillators localized in the prefrontal cortex area **(MEM)**, presumably mimicking the activity associated with the nucleus basalis magnocellularis **(OUT)**, presumably mimicking the striatal spiny neurons that by integrating a very large number of different inputs and responding selectively to particular reinforced patterns **(MOD)** that mimics the modulation of cortical and thalamic-induced activity of the striatal spiny neurons. The **MOD** block also modulates the threshold for coherent activity detection due to dopamine release from substantia niagra pars compacta

Schematic representation of the functional blocks of the SBF model

**Schematic representation of the functional blocks of the SBF model.** The oscillator block **OSC** contains _{osc} neural oscillators that constitute the time base for the entire interval timing network. The memory block **MEM** stores the criterion time, _{osc}. The decision and output block **OUT** compares the current state of the oscillators in **OSC**, **MOD** mimics the global effect of DA and ACh neuromodulators.

The oscillator block (OSC)

is composed of _{
osc
} neural oscillators with frequencies distributed over a range (_{1},_{2}) consistent with neurobiological observations _{
i
}, are equally spaced, i.e, _{
i
}=_{1}+_{2}−_{1})/_{
osc
}. **OSC** provides the underlying time base for the interval timing network. In the presence of noise, e.g., ionic channel noise _{
f
} of _{
osc
} frequencies, _{
i
} with a frequency variability _{
f
} that obeys a given probability density function _{
f
}. The output function is an average over all _{
f
} distributions of frequencies.

The memory block (MEM)

stores a criterion time value, _{
c
} of randomly distributing values _{
c
} according to a given probability density function _{
c
}. The output function averages over all _{
c
} randomly distributed values of the criterion time

The decision/output block (OUT)

relates the internal perception of time with external actions.

In order to implement the decision-making process ascribed to basal ganglia, we define a set of numbers (weights) that represent the state of each oscillator. The weight ^{
t
h
} neural oscillator from the **OSC** block at the reinforcement (criterion) time. Although it is not the only possibility, the “state” of the brain at the reinforcement time could be given, for example, by the phases or the amplitudes of all neural oscillators in **OSC**. The **OUT** block estimates the “closeness” between the current state of the brain represented by the running weights

The neuromodulator block (MOD)

mimics the experimentally observed effects of neuromodulators on interval timing. The actual mechanism implemented in this SBF model directly changes the firing frequency of all neurons in the **OSC** block proportional to the level of neuromodulator. In this implementation of the SBF model, we used the **MOD** block as a “start gun” that resets the **OSC** block at the beginning of each trial such that all neural oscillators state in phase. Elsewhere

The SBF model with cosine oscillators

In order to gain insight into the functionality of the SBF block model, we initially assumed that the time base is provided by cosine (phase) oscillators. A phase oscillator is a mathematical abstraction obtained by reducing a complex and detailed mathematical model of a biological neuron to a single parameter - the firing phase measured with respect to an arbitrary reference

In our implementation of the SBF model, the reference weights

where the sum is considered over all stored criteria

and found no significant difference in the properties of the output function.

In this implementation of the SBF model, **OUT** works as a phase detector, i.e., if the current vector of weights **OUT** block computes the current weights

Based on (3), we computed the absolute value of the cosine of the angle between

Results

Cosine oscillators with no variability

We gained significant insight into the dynamics of SBF model by assuming no noise (variability) in any of the model’s parameters. According to (1), the state of the **OSC** bock at the reinforcement time, i.e., the reference weights ^{
t
h
} phase (cosine) oscillator. According to (3), the output function of the SBF model with noiseless cosine oscillators is:

which becomes:

The output function (5) has two symmetric and strong peaks at _{
osc
}
_{1}+_{
osc
}
_{
osc
} as _{
osc
}
_{
osc
}
_{
osc
}, which shows that the

Normalized output functions for SBF model with cosine oscillators

**Normalized output functions for SBF model with cosine oscillators.** In the absence of any variability in the SBF model, the output functions are almost identical regardless the criterion time **(A)** and their widths are constant regardless the criterion times **(B)**. In the presence of uniformly distributed memory variability, the width of the output unction increases with the criterion time **(C)**. The width of the Gaussian envelope (dashed line in **C**) linearly increases with the criterion time both for uniform (solid rhombs in panel **D**) and normally-distributed criterion times (solid squares in panel **D**).

Cosine oscillators with arbitrary memory variability

As mentioned in the Introduction, biological noise is ubiquitous both as channel noise affecting the dynamics of individual oscillators

We only considered the physically realizable first term centered at

where _{
j
}=_{
j
}). In the presence of memory variability, the criterion time is a stochastic variable _{
j
}=_{
j
}) where _{
X
}(

The _{
z
} of the new stochastic variable _{
x
} of the criterion time _{
X
}(_{
Z
}(_{
X
}(_{
c
} stochastic variables with the _{
Z
}(

What about the time-scale invariance property? Is this feature of the output function still preserved regardless the

where the range (_{
min
},_{
max
}) depends on the type of _{
X
}(_{
min
}<_{
max
} such that (9) becomes:

To compute the width of the output function we introduced the dimensionless variable _{0}=(_{0}−

Using _{0}=

and _{0}=_{0} exists for Eq. (12), then the width _{0} increases linearly with the criterion time

We carried out numerical simulations using cosine model with _{
c
} different criterion times distributed around _{
c
} criteria are drawn from a uniform distribution centered on

The SBF with biophysically realistic model oscillators

Cosine oscillators were extensively used in numerical simulations of interval timing models with great success

ML oscillators with no variability

In the absence of any variability, our numerical results show that the width of the output function of the SBF model with ML oscillators does not change with criterion time, therefore, violating the scaling property. This finding is not surprising and it was predicted analytically in the case of cosine models. Since any periodic waveform, such as the action potential of an endogenously spiking neuron, can be decomposed in discrete cosine components, we conjectured that “no variability = no scalar property” based on the theoretical results obtained with cosine oscillators. We also noticed that the width of the output function decreases with the increase in the number of neural oscillators. Based on our cosine oscillator results, this observation is also predicable since the output function is the discrete Fourier transform of the reference weights vector

ML oscillators with arbitrary memory variability

The fact that noise, whether as channel noise

In order to maintain the parallel with the cosine (phase) model, we report here only the effect of memory variability on the standard deviation of the Gaussian fit of the output function generated by the SBF model with ML oscillators (see Figure ^{2}=0.342) for 0.1^{2}=0.789) for 1% variance (Figure ^{2}=0.898) for 10% memory variance (Figure _{
output
}∝_{
c
}

Normalized output functions for SBF model with ML oscillators

**Normalized output functions for SBF model with ML oscillators.** The width of the output function of the SBF model with 600 ML model neurons in the frequency range from 5.5 Hz to 11.5 Hz with normally distributed criterion times is insensitive to low 0.1**(A)** and 1% **(B)** levels of noise. The width of the output function shows linear increase with the criterion time for high noise levels, i.e., 10% **(C)** and panel **D**.

Discussion

Interval timing models vary largely with respect to the fundamental assumptions and the hypothesized mechanisms by which temporal processing is explained. In addition, interval timing model attempt explaining time-scale invariance, or drug effects differently. Among the most prominent models of interval timing we cite pacemaker/accumulator processes

By and large, to address time-scale invariance current behavioral theories assume convenient computations, rules, or coding schemes. Scalar timing is explained as either deriving from computation of ratios of durations

For example, Killeen and Taylor (1988) explained time-scale invariance of timing in terms of noisy information transfer during counting. Similarly, here, we explained time-scale invariance of timing in terms of noisy coincidence detection during timing. Our theoretical predictions based on an SBF model show that time-scale invariance emerges as the property of a (very) large and noisy network. Furthermore, our results regarding the effect of noise on interval timing support and extend the speculation

Conclusions

We investigated both analytically and numerically the properties of the output function generated by the SBF model and found that the output function mimics behavioral responses of animals performing peak interval procedures. We found analytically that, in the absence of any kind of variability in the parameters of the SBF model, the width of the output function only depends on the number of oscillators and the range of frequencies they cover. Therefore, in the absence of parameter variability the scalar property is violated.

We showed that if parameter variability is allowed, then the output function of the SBF model with cosine oscillators is always Gaussian, which is a consequence of the central limit theorem, regardless the

We also conjectured that the following two statements are always true in any noisy SBF implementation: (1) the output function is always Gaussian, which is a consequence of central limit theorem, and (2) the scalar property is valid regardless the

Appendix

Cosine model with no variances violates the scalar property

Close to the criterion time, c, only the fist term in (5) is significant. We used the least square fit method to approximate its envelope with a Gaussian centered on the criterion time. The output function becomes:

where _{
osc
}
_{1}/_{
osc
})_{
osc
}
_{
osc
}
_{0})=_{
osc
}
_{0}). The corresponding maximum values of the output function (13) are:

The pairs (_{0},_{0}) are determined by the number of oscillators _{
osc
} in the network and the range of frequencies covered. However, since there is no dependence of (_{0},_{0}) pair on the criterion time the output function is simply centered on _{
out
} of the output function envelope depends only on the range of oscillators’ frequencies _{1} and _{2}=_{
osc
}

Morris-Lecar model equations

We used a dimensionless, conductance-based, Morris-Lecar model

where _{1} is the membrane potential, _{2} is the slow potassium activation and all ionic currents are described by _{
x
}=_{
x
}(_{1}−_{
x
}), where _{
x
} is the conductance of the voltage gated channel _{
x
} is the corresponding reversal potential. In particular, the calcium current is _{
Ca
}=_{
Ca
}
_{
∞
}(_{1})(_{1}−_{
Ca
}), the potassium current is _{
K
}=_{
K
}
_{2}(_{1}−_{
K
}), and the leak current is _{
L
}=_{
L
}(_{1}−_{
L
}). The reversal potentials for calcium, potassium and leak currents are _{
Ca
}=1.0,_{
K
}=−0.7,_{
L
}=−0.5, respectively. The steady state activation function for calcium channels is _{
∞
}(_{1})=1+ tanh((_{1}−_{1})/_{2}))/2, where _{1}=−0.01,_{2}=0.15, the steady state activation function for potassium channels is _{
∞
}(_{1})=(1+ tanh((_{1}−_{3})/_{4}])/2 where _{3}=0.1,_{4}=0.145, the inverse time constant of potassium channels is _{0}(_{1})= cosh((_{1}−_{3})/_{4}/2), the potassium and leak conductances are _{
K
}=2.0,_{
L
}=0.5, respectively, and the

The two control parameters that can switch the ML model from a Type 1 excitable cell _{
Ca
} and the bias current _{0}. If _{
Ca
}=1.0 and 0.083<_{0}<0.242 the equations (15) describe what was classified by A.L. Hodgkin as Type 1 excitable cells. If _{
Ca
}=0.5 and 0.303<_{0}<0.138 the equations (15) describe a Type 2 excitable cells. In our simulations, we used a Type 2 ML model neuron that has a membrane potential shape very close to a cosine waveform.

Competing interests

The authors declare no disclosure of financial interests and potential conflict of interest.

Authors’ contributions

The analytical results regarding the Gaussian shape of the output function and the scalar property were obtained by SAO. CVB contributed to the implementation of the SBF model with cosine oscillators. SAO implemented the SBF model with ML model neurons and conducted all numerical simulations. Both authors contributed equally to the draft the manuscript. Both authors read and approved the final manuscript.

Acknowledgements

This research was supported by the CAREER award IOS 1054914 from the National Science Foundation to SAO and by the National Institutes of Health grants MH65561 and MH73057 to CVB.