Linear Algebra for Computation Neuroscience

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89: check that for t = 0 it indeed gives v(0), and that, for m < 0, as t → ∞ v → v FP d v(t) = 0. Thus, for where v FP = −h/m is the fixed point of Eq. 86, defined as the point where dt m < 0, Eq. 89 describes exponential decay with time constant 1/m from v(0) to v FP = −h/m. Now assume that M has a complete basis of N eigenvectors ei with eigenvalues λi . Express v and h in this basis: v(t) = i vi (t)ei , h(t) = i hi (t)ei , where vi (t) = eTi v(t) and hi (t) = eTi h(t). Then Eq. 85 becomes a set of N independent 1-dimensional equations, d vi (t) = λi vi + hi (t) dt for i = 0, .

J0 j0 The vector represents equal strengths of all synapses, while the vector repj0 −j0 resents equal strengths for all left-eye synapses, and equal and opposite strengths for all right-eye synapses. Thus, if the two eyes are correlated ( > 0), the sum of the two eyes’ strengths grows faster than the difference of their strengths; while if the two eyes are anticorrelated, the difference of the two eyes’ strengths grows faster than their sum, meaning that one eye’s strengths will grow and the other eye’s strengths will shrink.

That is, given any other complex number c2 = r2 eıθ2 , then cc2 = c2 c = rr2 eı(θ+θ2 ) . The complex numbers of the form eıθ — the complex numbers of modulus 1 — form a circle of radius 1 in the complex plane. As θ goes from zero to 2π, eıθ goes around this circle counterclockwise, beginning on the RE axis for θ = 0 and returning to the RE axis for θ = 2π. It will be critical to understand these numbers in order to understand the Fourier transform. Problem 29 Understanding the numbers eıθ : • Show that eıθ = 1, ı, −1, −ı for θ = 0, π/2, π, 3π/2 respectively.

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