Receptron

The '''receptron''' (short for "reservoir perceptron") is a neuromorphic data processing model — specifically neuromorphic computing — that generalizes the traditional perceptron, by incorporating non-linear interactions between inputs.<ref> {{Cite journal |last1=Mirigliano |first1=Matteo |last2=Paroli |first2=Bruno |last3=Martini |first3=Gianluca |last4=Fedrizzi |first4=Marco |last5=Falqui |first5=Andrea |last6=Casu |first6=Alberto |last7=Milani |first7=Paolo |date=2021-12-01 |title=A binary classifier based on a reconfigurable dense network of metallic nanojunctions |url=https://iopscience.iop.org/article/10.1088/2634-4386/ac29c9 |journal=Neuromorphic Computing and Engineering |volume=1 |issue=2 |pages=024007 |doi=10.1088/2634-4386/ac29c9 |issn=2634-4386|hdl=10754/671932 |hdl-access=free }} </ref><ref>{{Citation |last1=Paroli |first1=B. |title=The receptron is a nonlinear threshold logic gate with intrinsic multi-dimensional selective capabilities for analog inputs |date=2025-06-24 |arxiv=2506.19642|last2=Borghi |first2=F. |last3=Potenza |first3=M. A. C. |last4=Milani |first4=P.}} </ref><ref>{{Cite journal |last1=Perez |first1=Jake C. |last2=Shaheen |first2=Sean E. |date=August 2020 |title=Neuromorphic-based Boolean and reversible logic circuits from organic electrochemical transistors |url=http://link.springer.com/10.1557/mrs.2020.202 |journal=MRS Bulletin |language=en |volume=45 |issue=8 |pages=649–654 |doi=10.1557/mrs.2020.202 |bibcode=2020MRSBu..45..649P |issn=0883-7694|url-access=subscription }} </ref> Unlike classical perceptron, which rely on linearly independent weights, the receptron leverages complexity in physical substrates,<ref> {{Cite journal |last1=Stieg |first1=Adam Z. |last2=Avizienis |first2=Audrius V. |last3=Sillin |first3=Henry O. |last4=Martin-Olmos |first4=Cristina |last5=Aono |first5=Masakazu |last6=Gimzewski |first6=James K. |date=2012-01-10 |title=Emergent Criticality in Complex Turing B-Type Atomic Switch Networks |url=https://onlinelibrary.wiley.com/doi/10.1002/adma.201103053 |journal=Advanced Materials |language=en |volume=24 |issue=2 |pages=286–293 |doi=10.1002/adma.201103053 |pmid=22329003 |bibcode=2012AdM....24..286S |issn=0935-9648|url-access=subscription }} </ref> such as the electric conduction properties of nanostructured materials or optical speckle fields, to perform classification tasks.<ref>{{Cite journal |last1=Paroli |first1=B. |last2=Martini |first2=G. |last3=Potenza |first3=M. A. C. |last4=Siano |first4=M. |last5=Mirigliano |first5=M. |last6=Milani |first6=P. |date=2023-09-01 |title=Solving classification tasks by a receptron based on nonlinear optical speckle fields |url=https://www.sciencedirect.com/science/article/pii/S0893608023004203 |journal=Neural Networks |volume=166 |pages=634–644 |doi=10.1016/j.neunet.2023.08.001 |pmid=37604074 |issn=0893-6080 |archive-date=2024-04-18 |access-date=2025-09-03 |archive-url=https://web.archive.org/web/20240418205454/https://www.sciencedirect.com/science/article/pii/S0893608023004203 |url-status=live |hdl=2434/1026912 |hdl-access=free }} </ref><ref>{{Cite journal |last1=Iyer |first1=Prasad P. |last2=Bhatt |first2=Gaurang R. |last3=Desai |first3=Saaketh |last4=Fuller |first4=Elliot J. |last5=Teeter |first5=Corinne M. |last6=Léonard |first6=François |last7=Vineyard |first7=Craig M. |date=2025-08-08 |title=Is Computing with Light All You Need? A Perspective on Codesign for Optical Artificial Intelligence and Scientific Computing |url=https://advanced.onlinelibrary.wiley.com/doi/10.1002/aisy.202500371 |journal=Advanced Intelligent Systems |article-number=2500371 |language=en |doi=10.1002/aisy.202500371 |issn=2640-4567|doi-access=free }} </ref> The receptron bridges unconventional computing and neural network principles,<ref>{{Cite journal |last1=Frenkel |first1=Charlotte |last2=Bol |first2=David |last3=Indiveri |first3=Giacomo |date=June 2023 |title=Bottom-Up and Top-Down Approaches for the Design of Neuromorphic Processing Systems: Tradeoffs and Synergies Between Natural and Artificial Intelligence |journal=Proceedings of the IEEE |volume=111 |issue=6 |pages=623–652 |doi=10.1109/JPROC.2023.3273520 |issn=0018-9219}}</ref> enabling solutions that do not require the training approaches typical of artificial neural networks based on the perceptron model.<ref>{{Cite journal |last1=Barrows |first1=Frank |last2=Lin |first2=Jonathan |last3=Caravelli |first3=Francesco |last4=Chialvo |first4=Dante R. |date=July 2025 |title=Uncontrolled Learning: Codesign of Neuromorphic Hardware Topology for Neuromorphic Algorithms |url=https://advanced.onlinelibrary.wiley.com/doi/10.1002/aisy.202400739 |journal=Advanced Intelligent Systems |language=en |volume=7 |issue=7 |article-number=2400739 |doi=10.1002/aisy.202400739 |issn=2640-4567|doi-access=free }}</ref>

== Algorithm == The receptron is an algorithm for supervised learning of binary classifiers, so a classification algorithm that makes its predictions based on a predictor function, combining a set of weights with the feature vector.<ref>{{Cite journal |last1=Widrow |first1=B. |last2=Lehr |first2=M.A. |date=September 1990 |title=30 years of adaptive neural networks: perceptron, Madaline, and backpropagation |journal=Proceedings of the IEEE |volume=78 |issue=9 |pages=1415–1442 |doi=10.1109/5.58323 |bibcode=1990IEEEP..78.1415W }}</ref> The mathematical model is based on the sum of inputs with non-linear interactions:

<math>S = \sum_{k=1}^n x_j \widetilde{w}_j (\vec{x}) | S \in R</math> (1)

where <math>j \in [1, n]</math> and <math>\widetilde{w}_j </math> are non-linear weight functions depending on the inputs, <math>\vec{x}</math>. Nonlinearity will typically make the system extremely complex, and allowing for the solution of problems not solvable through the simpler rules of a linear system, such as the perceptron or McCulloch Pitts neurons, which is based on the sum of linearly independent weights:<ref>{{Citation |last1=Shukla |first1=Anupam |title=Artificial Neural Networks |date=2010 |work=Towards Hybrid and Adaptive Computing |volume=307 |pages=31–58 |url=http://link.springer.com/10.1007/978-3-642-14344-1_2 |access-date=2025-11-06 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-14344-1_2 |isbn=978-3-642-14343-4 |last2=Tiwari |first2=Ritu |last3=Kala |first3=Rahul|url-access=subscription }}</ref>

<math>S = \sum_{k=1}^n x_j w_j^p </math> (2)

where <math>w_j</math>are constant real values. A consequence of this simplicity is the limitation to linearly separable functions, which necessitates multi-layer architectures and training algorithms like backpropagation<ref>{{Cite journal |last=Goh |first=A.T.C. |date=January 1995 |title=Back-propagation neural networks for modeling complex systems |url=https://linkinghub.elsevier.com/retrieve/pii/095418109400011S |journal=Artificial Intelligence in Engineering |language=en |volume=9 |issue=3 |pages=143–151 |doi=10.1016/0954-1810(94)00011-S|url-access=subscription }}</ref>

As in the perceptron case,<ref>{{Cite journal |last=Block |first=H. D. |date=1962-01-01 |title=The Perceptron: A Model for Brain Functioning. I |url=https://link.aps.org/doi/10.1103/RevModPhys.34.123 |journal=Reviews of Modern Physics |language=en |volume=34 |issue=1 |pages=123–135 |doi=10.1103/RevModPhys.34.123 |bibcode=1962RvMP...34..123B |issn=0034-6861|url-access=subscription }}</ref> the summation in Eq. 1 origins the activation of the receptron output through the thresholding process,

<math>Y(x_1, ..., x_n) = \begin{cases} 1 & \text{if } S > \text{th} \\ 0 & \text{if } S \leq \text{th} \end{cases}</math> (3)

where th is a constant threshold parameter. Equation 3 can be written by using the Heaviside step function.

The weight functions <math>\widetilde{w} (\vec{x})</math> can be written with a finite number of parameters <math>w_{j_1...j_n}</math>, simplifying the model representation. One can Taylor-expand <math>\widetilde{w} (\vec{x})</math> and use the idempotency of Boolean variables <math>(x_j)^q = x_j \forall q \geq 1</math> such that <math>S' = b + \sum_{k=1}^n x_j \widetilde{w}_j (\vec{x})</math> can be written as

<math>S'(\vec{x}) = b + \sum_{j} w_j x_j + \sum_{j<k} w_{jk} x_j x_k + \sum_{j<k<l} w_{jkl} x_j x_k x_l + ... </math> (4)

where <math>w_{j_1...j_n}</math> are independent parameters that can be seen as the components of a tensor <math>W</math> (“weight tensor”) of rank <math>n</math> and type <math>(n,0)</math>.

The sum in Eq. [3] reduces to the perceptron case when off-diagonal terms of <math>W</math> vanish. If one considers <math>n=2</math>, one gets:

<math>S'(\vec{x}) = b + x_1 w_{11} + x_2 w_{22} + x_1 x_2 w_{12}</math> (5)

in the perceptron case, the vanishing of <math>w_{12}</math> implies linearity <math>S(1,1)=S(0,1) + S(1,0)</math>. In the receptron case <math>S(1,1) \neq S(0,1) + S(1,0)</math>, meaning that the superposition principle is no longer valid, the latter terms being responsible of the more complex non-linear interaction between the inputs.

== Design and implementations ==

=== 1. Electrical Receptron === Substrate: Nanostructured and nanocomposite films (Au, Pt, Zr Au/Zr). These films form disordered networks of nanoparticles with resistive switching and non-linear electrical conduction.

=== 2. Optical Receptron === Substrate: Optical speckle fields generated by random interference of light emerging from a disordered medium illuminated by a laser or coherent radiation.<ref>{{Cite journal |last1=Paroli |first1=Bruno |last2=Malfer |first2=Alessandro |last3=Potenza |first3=Marco A.C. |last4=Siano |first4=Mirko |last5=Milani |first5=Paolo |date=2025-08-21 |title=Binary Pattern Classification with a Photonic Neuromorphic Device Based on Optical Receptrons |url=https://onlinelibrary.wiley.com/doi/10.1002/lpor.202500970 |journal=Laser & Photonics Reviews |article-number=e00970 |language=en |doi=10.1002/lpor.202500970 |issn=1863-8880|doi-access=free |hdl=2434/1208162 |hdl-access=free }}</ref>

== Key features == Physical Substrate Computing: The receptron does not require digital training; instead, it exploits the natural complexity of materials (e.g., nanowire networks, diffractive media) to perform computations.

Non-Linear Separability: Unlike traditional perceptrons, which fail on problems like the XOR function, the receptron can solve such tasks due to its inherent non-linearity.

Training-Free Operation: Classification is achieved through the physical system's response rather than iterative weight adjustments, reducing computational overhead.

== References == {{Reflist}}

Category:Artificial neural networks