The Basal Ganglia System as an Engine for Exploration

Abstract

One of the earliest attempts at building a theory of the basal ganglia (BG) is based on the clinical findings that lesions to the direct and indirect pathways of the BG produce quite opposite motor manifestations (Albin et al., in Trends Neurosci 12(10):366–375, 1989). While lesions of the direct pathway (DP), affecting particularly the projections from the striatum to GPi, are associated with hypokinetic disorders (distinguished by a paucity of movement), lesions of the indirect pathway (IP) produce hyperkinetic disorders, such as chorea and tremor. In this chapter, we argue that describing the two BG pathways as having mutually opponent actions has limitations. We argue that the BG indirect pathway also plays a role in exploration. We should evidence from various motor learning and decision-making tasks that exploration is a necessary process in various behavioral processes. Importantly, we use the exploration mechanism explained here to simulate various processes of the basal ganglia which we discuss in the following chapters.

Appendix

The system of equations for single oscillator is given by,

$$ \frac{{{\text{d}}x}}{{{\text{d}}t}} = - x + v - s + I $$

(i)

$$ v = { \tan }\,{{h}}\left( {\lambda x} \right) $$

(ii)

$$ \frac{{{\text{d}}s}}{{{\text{d}}t}} = - s + v $$

(iii)

Differentiating (i)

$$ \frac{{{\text{d}}^{2} x}}{{{\text{d}}^{2} t}} = - \frac{{{\text{d}}x}}{{{\text{d}}t}} + \lambda \,\sec \,h^{2} \left( {\lambda x} \right)\frac{{{\text{d}}x}}{{{\text{d}}t}} - \frac{{{\text{d}}s}}{{{\text{d}}t}} $$

(iv)

Substituting (ii) and (iii) in (iv)

$$ \frac{{{\text{d}}^{2} x}}{{{\text{d}}^{2} t}} = - \frac{{{\text{d}}x}}{{{\text{d}}t}} + \lambda \,\sec \,h^{2} \left( {\lambda x} \right)\frac{{{\text{d}}x}}{{{\text{d}}t}} - \left( { - s + \tan \,h\left( {\lambda x} \right)} \right) $$

(v)

Substituting (i) and (ii) in (v)

$$ \frac{{{\text{d}}^{2} x}}{{{\text{d}}^{2} t}} = - \frac{{{\text{d}}x}}{{{\text{d}}t}} + \lambda \,\sec \,h^{2} \left( {\lambda x} \right)\frac{{{\text{d}}x}}{{{\text{d}}t}} - \left( {\frac{{{\text{d}}x}}{{{\text{d}}t}} + x - v - I + \tan \,h\left( {\lambda x} \right)} \right) $$

on rearranging

$$ \frac{{{\text{d}}^{2} x}}{{{\text{d}}^{2} t}} + \frac{{{\text{d}}x}}{{{\text{d}}t}}\left( {2 - \lambda \,\sec \,h^{2} \left( {\lambda x} \right)} \right) + \left( {x - I} \right) = 0 $$

(vi)

is similar to Lienard’s equation $\frac{{{\text{d}}^{2} x}}{{{\text{d}}^{2} t}} + \frac{{{\text{d}}x}}{{{\text{d}}t}}f\left( x \right) + g\left( x \right) = 0$ where $f\left( x \right) = 2 - \lambda \,\sec \,h^{2} \left( {\lambda x} \right)$, and $g(x) = x - I$.

Checking for the Lienard’s conditions, let us assume I = 0.

• 1. Both $f(x)$ and $g(x)$ are continuously differentiable for all $x$.

• 2. $g( - x) = - g(x)$ for all $x$ (i.e., $g(x)$ is an odd function).

• 3. $g(x) > 0\;{\text{for}}\;x > 0$.

• 4. For all $x$(i.e., $f(x)$ is an even function); The odd function $F(x) = \int\nolimits_{0}^{x} {f(u)du}$ = $2x - \tan \,h\left( {\lambda x} \right)$ has exactly one positive zero at $x = x_{o}$, is negative for $0 < x < x_{o}$, is positive and non-decreasing for $x > x_{o}$, and $F(x) \to \infty$ as $x \to \infty$ (one can estimate $x_{o}$ from graph of $F(x)$). So the system has a unique stable limit cycle surrounding the origin in the phase plane.