# Stimulus–response model

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The stimulus–response model is a characterization of a statistical unit (such as a neuron). The model allows the prediction of a quantitative response to a quantitative stimulus, for example one administered by a researcher. In psychology, stimulus response theory concerns forms of classical conditioning in which a stimulus becomes paired response in a subject's mind.[1]

## Fields of application

Stimulus–response models are applied in international relations,[2] psychology,[3] risk assessment,[4] neuroscience,[5] neurally-inspired system design,[6] and many other fields.

Pharmacological dose response relationships are an application of stimulus-response models.

## Mathematical formulation

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:[7]

${\displaystyle E(Y)=f(x)}$

A common simplification assumed for such functions is linear, thus we expect to see a relationship like

${\displaystyle E(Y)=\alpha +\beta x.}$

Statistical theory for linear models has been well developed for more than fifty years, and a standard form of analysis called linear regression has been developed.

### 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.[8] Empirical models based on nonlinear regression are usually preferred over the use of some transformation of the data that linearizes the stimulus-response relationship.[9]

One example of a logit model for the probability of a response to the real input (stimulus) ${\displaystyle x}$, (${\displaystyle x\in \mathbb {R} }$) is

${\displaystyle p(x)={\frac {1}{1+e^{-(\beta _{0}+\beta _{1}x)}}}}$

where ${\displaystyle \beta _{0},\beta _{1}}$ are the parameters of the function.

Conversely, a Probit model would be of the form

${\displaystyle p(x)=\Phi (\beta _{0}+\beta _{1}x)}$

where ${\displaystyle \Phi (x)}$ is the cumulative distribution function of the normal distribution.

#### Hill equation

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.[10] The Hill equation is important in the construction of dose-response curves. The Hill equation is the following formula, where ${\displaystyle E}$ is the magnitude of the response, ${\displaystyle {\ce {[A]}}}$ is the drug concentration (or equivalently, stimulus intensity), ${\displaystyle \mathrm {EC} _{50}}$ is the drug concentration that produces a half-maximal response and ${\displaystyle n}$ is the Hill coefficient.

${\displaystyle {\frac {E}{E_{\mathrm {max} }}}={\frac {1}{1+\left({\frac {\mathrm {EC} _{50}}{[A]}}\right)^{n}}}}$[10]

Note that the Hill equation rearranges to a logistic function with respect to the logarithm of the dose (similar to a logit model).

## References

1. ^ Biscontini, Tyler J. "What is the stimulus-response theory in psychology?".
2. ^ Greg Cashman (2000). "International Interaction: Stimulus–Response Theory and Arms Races". What causes war?: an introduction to theories of international conflict. Lexington Books. pp. 160–192. ISBN 978-0-7391-0112-4.
3. ^ Stephen P. Kachmar and Kimberly Blair (2007). "Counseling Across the Life Span". In Jocelyn Gregoire and Christin Jungers (ed.). The Counselor's Companion: What Every Beginning Counselor Needs to Know. Routledge. p. 143. ISBN 978-0-8058-5684-2.
4. ^ Walter W. Piegorsch and A. John Bailer (2005). "Quantitative Risk Assessment with Stimulus–Response Data". Analyzing environmental data. John Wiley and Sons. pp. 171–214. ISBN 978-0-470-84836-4.
5. ^ Geoffrey W. Hoffmann (1988). "Neurons with hysteresis?". In Rodney Cotterill (ed.). Computer simulation in brain science. Cambridge University Press. pp. 74–87. ISBN 978-0-521-34179-0.
6. ^ Teodor Rus (1993). Systems methodology for software. World Scientific. p. 12. ISBN 978-981-02-1254-4.
7. ^ Meyer, A. F., Williamson, R. S., Linden, J. F., & Sahani, M. (2017). Models of neuronal stimulus-response functions: elaboration, estimation, and evaluation. Frontiers in systems neuroscience, 10, 109.
8. ^ Hamilton, MA; Russo, RC; Thurston, RV (1977). "Trimmed Spearman-Karber method for estimating median lethal concentrations in toxicity bioassays". Environmental Science & Technology. 11 (7): 714–9. Bibcode:1977EnST...11..714H. doi:10.1021/es60130a004.
9. ^ Bates, Douglas M.; Watts, Donald G. (1988). Nonlinear Regression Analysis and its Applications. Wiley. p. 365. ISBN 9780471816430.
10. ^ a b Neubig, Richard R. (2003). "International Union of Pharmacology Committee on Receptor Nomenclature and Drug Classification. XXXVIII. Update on Terms and Symbols in Quantitative Pharmacology" (PDF). Pharmacological Reviews. 55 (4): 597–606. doi:10.1124/pr.55.4.4. PMID 14657418. S2CID 1729572.