The stimulus–response model is a characterization of a statistical unit (such as a neuron) as a black box model, predicting a quantitative response to a quantitative stimulus, for example one administered by a researcher.
Fields of application
Pharmacological dose response relationships are an application of stimulus-response models.
The object of a stimulus–response model is to establish a mathematical function that describes the relation f between the stimulus x and the expected value (or other measure of location) of the response Y:
A common simplification assumed for such functions is linear, thus we expect to see a relationship like
Bounded Response Functions
Since many types of response have inherent physical limitations (e.g. minimal maximal muscle contraction), it is often applicable to use a bounded function (such as the logistic function) to model the response. Similarly, a linear response function may be unrealistic as it would imply arbitrarily large responses. For binary dependent variables, statistical analysis with regression methods such as the probit model or logit model, or other methods such as the Spearman-Karber method. Empirical models based on nonlinear regression are usually preferred over the use of some transformation of the data that linearizes the stimulus-response relationship.
One example of a logit model for the probability of a response to the real input (stimulus) , () is
where are the parameters of the function.
Conversely, a Probit model would be of the form
In biochemistry and pharmacology, the Hill equation refers to two closely related equations, one of which describes the response (the physiological output of the system, such as muscle contraction) to Drug or Toxin, as a function of the drug's concentration. The Hill equation is important in the construction of dose-response curves. The Hill equation is the following formula, where is the magnitude of the response, is the drug concentration (or equivalently, stimulus intensity), is the drug concentration that produces a half-maximal response and is the Hill coefficient.
Note that the Hill equation rearranges to a logistic function with respect to the logarithm of the dose (similar to a logit model).
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