Dynamical Systems in Neuroscience:

326 Simple ModelsExercises1. (Integrate-and-fire network) The simplest implementation of a pulse-coupled integrate-and-fireneural network has the form˙v i = b i − v i + ∑ j≠ic ij δ(t − t j ) ,where t j is the moment of firing of the jth neuron, i.e., the moment v j (t j ) =1. Thus, whenever the jth neuron fires, the membrane potentials of the otherneurons are adjusted instantaneously by c ij , i ≠ j. Show that the same initialconditions may result in different solutions, depending on the implementationdetails.2. (Latham et al. 2000) Determine the relationship between the normal form forsaddle-node bifurcation (6.2) and the equation˙V = a(V − V rest )(V − V thresh ) .3. Show that the period of oscillations in the quadratic integrate-and-fire model(8.2) isT = √ 1 (atan v peak√ − atan v )reset√b b bwhen b > 0.4. Show that the period of oscillations in the quadratic integrate-and-fire model(8.2) with v peak = 1 is(T = 12 √ ln 1 − √ |b||b| 1 + √ |b| − ln v reset − √ )|b|v reset + √ |b|when b < 0 and v reset > √ |b|.5. Justify the bifurcation diagram in Fig. 8.3.6. Brizzi et al. (2004) have shown that shunting inhibition of cat motoneurons raisesthe firing threshold, rheobase current, and shifts the F-I curve to the right withoutchanging the shape of the curve. Use the quadratic integrate-and-fire model toexplain the effect. (Hint: Consider ˙v = b − gv + v 2 with g ≥ 0, v reset = −∞, andv peak = +∞.)7. (Theta neuron) Determine when the quadratic integrate-and-fire neuron (8.2) isequivalent to the theta neuron˙ϑ = (1 − cos ϑ) + (1 + cos ϑ)r , (8.8)where r is the bifurcation parameter and ϑ ∈ [−π, π] is a phase variable on theunit circle.