Receptron

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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.[1][2][3] Unlike classical perceptron, which rely on linearly independent weights, the receptron leverages complexity in physical substrates,[4] such as the electric conduction properties of nanostructured materials or optical speckle fields, to perform classification tasks.[5][6] The receptron bridges unconventional computing and neural network principles,[7] enabling solutions that do not require the training approaches typical of artificial neural networks based on the perceptron model.[8]

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.[9] The mathematical model is based on the sum of inputs with non-linear interactions:

S = ∑ k = 1 n x j w ~ j ( x → ) | S ∈ R {\displaystyle S=\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})|S\in R} {\displaystyle S=\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})|S\in R}      (1)

where j ∈ [ 1 , n ] {\displaystyle j\in [1,n]} {\displaystyle j\in [1,n]} and w ~ j {\displaystyle {\widetilde {w}}_{j}} {\displaystyle {\widetilde {w}}_{j}} are non-linear weight functions depending on the inputs, x → {\displaystyle {\vec {x}}} {\displaystyle {\vec {x}}}. 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:[10]

S = ∑ k = 1 n x j w j p {\displaystyle S=\sum _{k=1}^{n}x_{j}w_{j}^{p}} {\displaystyle S=\sum _{k=1}^{n}x_{j}w_{j}^{p}}     (2)

where w j {\displaystyle w_{j}} {\displaystyle w_{j}}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[11]

As in the perceptron case,[12] the summation in Eq. 1 origins the activation of the receptron output through the thresholding process,

Y ( x 1 , . . . , x n ) = { 1 if  S > th 0 if  S ≤ th {\displaystyle Y(x_{1},...,x_{n})={\begin{cases}1&{\text{if }}S>{\text{th}}\\0&{\text{if }}S\leq {\text{th}}\end{cases}}} {\displaystyle Y(x_{1},...,x_{n})={\begin{cases}1&{\text{if }}S>{\text{th}}\\0&{\text{if }}S\leq {\text{th}}\end{cases}}} (3)

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

The weight functions w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} {\displaystyle {\widetilde {w}}({\vec {x}})} can be written with a finite number of parameters w j 1 . . . j n {\displaystyle w_{j_{1}...j_{n}}} {\displaystyle w_{j_{1}...j_{n}}}, simplifying the model representation. One can Taylor-expand w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} {\displaystyle {\widetilde {w}}({\vec {x}})} and use the idempotency of Boolean variables ( x j ) q = x j ∀ q ≥ 1 {\displaystyle (x_{j})^{q}=x_{j}\forall q\geq 1} {\displaystyle (x_{j})^{q}=x_{j}\forall q\geq 1} such that S ′ = b + ∑ k = 1 n x j w ~ j ( x → ) {\displaystyle S'=b+\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})} {\displaystyle S'=b+\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})} can be written as

S ′ ( x → ) = b + ∑ j w j x j + ∑ j < k w j k x j x k + ∑ j < k < l w j k l x j x k x l + . . . {\displaystyle 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}+...} {\displaystyle 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}+...} (4)

where w j 1 . . . j n {\displaystyle w_{j_{1}...j_{n}}} {\displaystyle w_{j_{1}...j_{n}}} are independent parameters that can be seen as the components of a tensor W {\displaystyle W} {\displaystyle W} (“weight tensor”) of rank n {\displaystyle n} {\displaystyle n} and type ( n , 0 ) {\displaystyle (n,0)} {\displaystyle (n,0)}.

The sum in Eq. [3] reduces to the perceptron case when off-diagonal terms of W {\displaystyle W} {\displaystyle W} vanish. If one considers n = 2 {\displaystyle n=2} {\displaystyle n=2}, one gets:

S ′ ( x → ) = b + x 1 w 11 + x 2 w 22 + x 1 x 2 w 12 {\displaystyle S'({\vec {x}})=b+x_{1}w_{11}+x_{2}w_{22}+x_{1}x_{2}w_{12}} {\displaystyle S'({\vec {x}})=b+x_{1}w_{11}+x_{2}w_{22}+x_{1}x_{2}w_{12}} (5)

in the perceptron case, the vanishing of w 12 {\displaystyle w_{12}} {\displaystyle w_{12}} implies linearity S ( 1 , 1 ) = S ( 0 , 1 ) + S ( 1 , 0 ) {\displaystyle S(1,1)=S(0,1)+S(1,0)} {\displaystyle S(1,1)=S(0,1)+S(1,0)}. In the receptron case S ( 1 , 1 ) ≠ S ( 0 , 1 ) + S ( 1 , 0 ) {\displaystyle S(1,1)\neq S(0,1)+S(1,0)} {\displaystyle S(1,1)\neq S(0,1)+S(1,0)}, 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.[13]

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

  1. Mirigliano, Matteo; Paroli, Bruno; Martini, Gianluca; Fedrizzi, Marco; Falqui, Andrea; Casu, Alberto; Milani, Paolo (2021-12-01). "A binary classifier based on a reconfigurable dense network of metallic nanojunctions". Neuromorphic Computing and Engineering. 1 (2): 024007. doi:10.1088/2634-4386/ac29c9. hdl:10754/671932. ISSN 2634-4386.
  2. Paroli, B.; Borghi, F.; Potenza, M. A. C.; Milani, P. (2025-06-24), The receptron is a nonlinear threshold logic gate with intrinsic multi-dimensional selective capabilities for analog inputs, arXiv:2506.19642
  3. Perez, Jake C.; Shaheen, Sean E. (August 2020). "Neuromorphic-based Boolean and reversible logic circuits from organic electrochemical transistors". MRS Bulletin. 45 (8): 649–654. Bibcode:2020MRSBu..45..649P. doi:10.1557/mrs.2020.202. ISSN 0883-7694.
  4. Stieg, Adam Z.; Avizienis, Audrius V.; Sillin, Henry O.; Martin-Olmos, Cristina; Aono, Masakazu; Gimzewski, James K. (2012-01-10). "Emergent Criticality in Complex Turing B-Type Atomic Switch Networks". Advanced Materials. 24 (2): 286–293. Bibcode:2012AdM....24..286S. doi:10.1002/adma.201103053. ISSN 0935-9648. PMID 22329003.
  5. Paroli, B.; Martini, G.; Potenza, M. A. C.; Siano, M.; Mirigliano, M.; Milani, P. (2023-09-01). "Solving classification tasks by a receptron based on nonlinear optical speckle fields". Neural Networks. 166: 634–644. doi:10.1016/j.neunet.2023.08.001. hdl:2434/1026912. ISSN 0893-6080. PMID 37604074. Archived from the original on 2024-04-18. Retrieved 2025-09-03.
  6. Iyer, Prasad P.; Bhatt, Gaurang R.; Desai, Saaketh; Fuller, Elliot J.; Teeter, Corinne M.; Léonard, François; Vineyard, Craig M. (2025-08-08). "Is Computing with Light All You Need? A Perspective on Codesign for Optical Artificial Intelligence and Scientific Computing". Advanced Intelligent Systems 2500371. doi:10.1002/aisy.202500371. ISSN 2640-4567.
  7. Frenkel, Charlotte; Bol, David; Indiveri, Giacomo (June 2023). "Bottom-Up and Top-Down Approaches for the Design of Neuromorphic Processing Systems: Tradeoffs and Synergies Between Natural and Artificial Intelligence". Proceedings of the IEEE. 111 (6): 623–652. doi:10.1109/JPROC.2023.3273520. ISSN 0018-9219.
  8. Barrows, Frank; Lin, Jonathan; Caravelli, Francesco; Chialvo, Dante R. (July 2025). "Uncontrolled Learning: Codesign of Neuromorphic Hardware Topology for Neuromorphic Algorithms". Advanced Intelligent Systems. 7 (7) 2400739. doi:10.1002/aisy.202400739. ISSN 2640-4567.
  9. Widrow, B.; Lehr, M.A. (September 1990). "30 years of adaptive neural networks: perceptron, Madaline, and backpropagation". Proceedings of the IEEE. 78 (9): 1415–1442. Bibcode:1990IEEEP..78.1415W. doi:10.1109/5.58323.
  10. Shukla, Anupam; Tiwari, Ritu; Kala, Rahul (2010), "Artificial Neural Networks", Towards Hybrid and Adaptive Computing, vol. 307, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 31–58, doi:10.1007/978-3-642-14344-1_2, ISBN 978-3-642-14343-4, retrieved 2025-11-06
  11. Goh, A.T.C. (January 1995). "Back-propagation neural networks for modeling complex systems". Artificial Intelligence in Engineering. 9 (3): 143–151. doi:10.1016/0954-1810(94)00011-S.
  12. Block, H. D. (1962-01-01). "The Perceptron: A Model for Brain Functioning. I". Reviews of Modern Physics. 34 (1): 123–135. Bibcode:1962RvMP...34..123B. doi:10.1103/RevModPhys.34.123. ISSN 0034-6861.
  13. Paroli, Bruno; Malfer, Alessandro; Potenza, Marco A.C.; Siano, Mirko; Milani, Paolo (2025-08-21). "Binary Pattern Classification with a Photonic Neuromorphic Device Based on Optical Receptrons". Laser & Photonics Reviews e00970. doi:10.1002/lpor.202500970. hdl:2434/1208162. ISSN 1863-8880.