Abstract
In experiments described in [143], fast-firing inhibitory interneurons in the barrel cortex of mice, the part of the mouse brain that processes input from the whiskers, were driven to synchronize at 40 Hz, using optogenetic techniques. (In general, optogenetic techniques involve genetically sensitizing neurons to light, then using light to control them.) Figure 35.1 is a reproduction of Fig. 3A of [143]. The figure shows LFP recordings from barrel cortex during 40 Hz optogenetic stimulation of the fast-firing inhibitory interneurons. The central finding of [143] was that making the activity of fast-firing inhibitory interneurons rhythmic at 40 Hz improved the ability of the mice to perceive certain whisker deflections.
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Notes
- 1.
Because the integrand is smooth and periodic, and the integral extends over one period, the trapezoid method for evaluating integrals is of infinite order of accuracy here [78].
- 2.
For any function f = f(t), t ≥ 0, the notation “supt≥0 f(t)” denotes the supremum of f(t) over t ≥ 0, i.e., the lowest upper bound, the smallest A ∈ (−∞,∞] such that f(t) ≤ A for all t ≥ 0.
- 3.
Here is a completely precise way of reasoning. Suppose \(\overline{v}(t_{0}) >\hat{ v}\). Since \(\overline{v}\) is continuous, there is an ε > 0 with \(\overline{v}(t) >\hat{ v}\) for all t ∈ (t 0 −ε, t 0). We choose the largest possible such ε. Therefore \(\overline{v}(t_{0}-\epsilon ) =\hat{ v}\). The mean value theorem now implies that there is a time t strictly between t 0 −ε and t 0 at which \(d\overline{v}/dt > 0\). This is a contradiction because at this time, \(\overline{v} >\hat{ v}\), and we showed that \(\overline{v} >\hat{ v}\) implies \(d\overline{v}/dt < 0\).
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Börgers, C. (2017). Rhythmic vs. Tonic Inhibition. In: An Introduction to Modeling Neuronal Dynamics. Texts in Applied Mathematics, vol 66. Springer, Cham. https://doi.org/10.1007/978-3-319-51171-9_35
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