In this activity students will study a linear, first order, one-dimensional ordinary differential equation (ODE) and learn how it can be used to understand basics of neural dynamics. The modeling framework is known in the mathematical neuroscience literature as the ``integrate-and-fire'' neuron. The form of the model is equivalent to the linear ODE used to describe the RC circuit. This activity assumes familiarity with solving linear first order ODEs, for instance using the integrating factor method. Students are asked to set up and solve a homogeneous version of the equation and a nonhomogeneous version (constant forcing term). Students are asked to set up ODEs (select initial values), solve the ODEs, and perform some related algebraic calculations. Throughout, students are asked to interpret their results in light of biological experiments and biological terms that are explained in the activity. There is a focus throughout on solving the ODE in the presence of unspecified parameters and interpreting how parameter values may affect response characteristics of biological neurons. Outcomes of the project include: improved skill setting up and solving linear first order ODEs (including with unspecified parameters) and demonstrating how mathematical models can improve understanding of dynamic biological systems. There is no requirement or expectation that students have experience with biology or neuroscience.
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