Spiking Neural Network
Spiking neural network

							

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The insect is controlled by a spiking neural network to find a target in an unknown terrain.
Spiking neural networks (SNNs) are artificial neural networks (ANN) that more closely mimic natural neural networks.[1] These models leverage timing of discrete spikes as the main information carrier.[2]
In addition to neuronal and synaptic state, SNNs incorporate the concept of time into their operating model. The idea is that neurons in the SNN do not transmit information at each propagation cycle (as it happens with typical multi-layer perceptron networks), but rather transmit information only when a membrane potential—an intrinsic quality of the neuron related to its membrane electrical charge—reaches a specific value, called the threshold. When the membrane potential reaches the threshold, the neuron fires, and generates a signal that travels to other neurons which, in turn, increase or decrease their potentials in response to this signal. A neuron model that fires at the moment of threshold crossing is also called a spiking neuron model.[3]
Although it was previously believed that the brain encoded information through spike rates, which can be considered as the analogue variable output of a traditional ANN,[4] research in the field of neurobiology has indicated that high speed processing cannot solely be performed through a rate based scheme. For example humans can perform an image recognition task at rate requiring no more than 10ms of processing time per neuron through the successive layers (going from the retina to the temporal lobe). This time window is too short for a rate based encoding. The precise spike timings in a small set of spiking neurons also has a higher information coding capacity compared with a rate based approach.[5]
The most prominent spiking neuron model is the leaky integrate-and-fire model.[6] In the integrate-and-fire model, the momentary activation level (modeled as a differential equation) is normally considered to be the neuron's state, with incoming spikes pushing this value higher or lower, until the state eventually either decays or—if the firing threshold is reached—the neuron fires. After firing, the state variable is reset to a lower value.
Various decoding methods exist for interpreting the outgoing spike train as a real-value number, relying on either the frequency of spikes (rate-code), the time-to-first-spike after stimulation, or the interval between spikes.




Pulsed neuron model
Artificial synapses based on FTJs
Many multi-layer artificial neural networks are fully connected, receiving input from every neuron in the previous layer and signalling every neuron in the subsequent layer. Although these networks have achieved breakthroughs in many fields, they are biologically inaccurate and do not mimic the operation mechanism of neurons in the brain of a living thing.[citation needed]


The biologically inspired Hodgkin–Huxley model of a spiking neuron was proposed  in 1952. This model describes how action potentials are initiated and propagated. Communication between neurons, which requires the exchange of chemical neurotransmitters in the synaptic gap, is described in various models, such as the integrate-and-fire model, FitzHugh–Nagumo model (1961–1962), and Hindmarsh–Rose model (1984). The leaky integrate-and-fire model (or a derivative) is commonly used as it is easier to compute than the Hodgkin–Huxley model.[7]
While the notion of an artificial spiking neural network became very popular only during the first quarter of the twenty-first century, [8][9][10] there are a number of studies between 1980 and 1995 that supported the concept and in which the first models of this type of artificial neural networks appeared to simulate non-algorithmic intelligent information processing systems.[11][12][13] However, the very notion of the spiking neural network as a mathematical model had already been worked on in the early 1970s.[14]



Information in the brain is represented as action potentials (neuron spikes), which may be grouped into spike trains or even coordinated waves of brain activity. A fundamental question of neuroscience is to determine whether neurons communicate by a rate or temporal code.[15] Temporal coding suggests that a single spiking neuron can replace hundreds of hidden units on a sigmoidal neural net.[1]
An SNN computes in the continuous rather than the discrete domain. The idea is that neurons may not test for activation in every iteration of propagation (as is the case in a typical multilayer perceptron network), but only when their membrane potentials reach a certain value. When a neuron is activated, it produces a signal that is passed to connected neurons, raising or lowering their membrane potential.
In a spiking neural network, a neuron's current state is defined as its membrane potential (possibly modeled as a differential equation).[16] An input pulse causes the membrane potential to rise for a period of time and then gradually decline. Encoding schemes have been constructed to interpret these pulse sequences as a number, taking into account both pulse frequency and pulse interval. A neural network model based on pulse generation time can be established.[17] Using the exact time of pulse occurrence, a neural network can employ more information and offer better computing properties.[18]
The SNN approach produces a continuous output instead of the binary output of traditional artificial neural networks (ANNs). Pulse trains are not easily interpretable, hence the need for encoding schemes as above. However, a pulse train representation may be more suited for processing spatiotemporal data (or continual real-world sensory data classification).[19] SNNs consider space by connecting neurons only to nearby neurons so that they process input blocks separately (similar to CNN using filters). They consider time by encoding information as pulse trains so as not to lose information in a binary encoding. This avoids the additional complexity of a recurrent neural network (RNN). It turns out that impulse neurons are more powerful computational units than traditional artificial neurons.[20]
SNNs are theoretically more powerful than so called "second-generation networks" defined in[20] as "[ANNs] based on computational units that apply activation function with a continuous set of possible output values to a weighted sum (or polynomial) of the inputs; however, SNN training issues and hardware requirements limit their use. Although unsupervised biologically inspired learning methods are available such as Hebbian learning and STDP, no effective supervised training method is suitable for SNNs that can provide better performance than second-generation networks.[20] Spike-based activation of SNNs is not differentiable thus making it hard to develop gradient descent based training methods to perform error backpropagation.
SNNs have much larger computational costs for simulating realistic neural models than traditional ANNs.[21]
Pulse-coupled neural networks (PCNN) are often confused with SNNs. A PCNN can be seen as a kind of SNN.
Currently there are a few challenges when using SNNs that researchers are actively working on. The first challenge concerns the nondifferentiability of the spiking nonlinearity. The expressions for both the forward- and backward-learning methods contain the derivative of the neural activation function which is non-differentiable because neuron's output is either 1 when it spikes, and 0 otherwise. This all-or-nothing behavior of the binary spiking nonlinearity stops gradients from “flowing” and makes LIF neurons unsuitable for gradient-based optimization. The second challenge concerns the implementation of the optimization algorithm itself. Standard BP can be expensive in terms of computation, memory, and communication and may be poorly suited to the constraints dictated by the hardware that implements it (e.g., a computer, brain, or neuromorphic device).[22] Regarding the first challenge there are several approaches to resolving it. A few of them are:

resorting to entirely biologically inspired local learning rules for the hidden units
translating conventionally trained “rate-based” NNs to SNNs
smoothing the network model to be continuously differentiable
defining an SG (Surrogate Gradient) as a continuous relaxation of the real gradients
In the development of SNNs, incorporating additional neuron dynamics like Spike Frequency Adaptation (SFA) into neuron models marks a notable advance, enhancing both efficiency and computational power.[6][23] These neurons stand in between biological complexity and computational complexity.[24] Originating from biological insights, SFA offers significant computational benefits by reducing power usage through efficient coding,[25] especially in cases of repetitive or intense stimuli. This adaptation improves signal clarity against background noise and introduces an elementary short-term memory at the neuron level, which in turn, refines the accuracy and efficiency of information processing.[26] Recently, this phenomenon was mostly achieved using compartmental neuron models. The simpler versions are of neuron models with adaptive thresholds, an indirect way of achieving SFA. It equips SNNs with improved learning capabilities, even with constrained synaptic plasticity, and elevates computational efficiency.[27][28] This feature lessens the demand on network layers by decreasing the need for spike processing, thus cutting down on computational load and memory access time—essential aspects of neural computation. Moreover, SNNs utilizing neurons capable of SFA achieve levels of accuracy that rival those of conventional artificial neural networks, including those based on long short-term memory models,[29][30] while also requiring fewer neurons for comparable computational tasks. This efficiency not only streamlines the computational workflow but also conserves space and energy, offering a pragmatic step forward in the practical application of SNNs for complex computing tasks while maintaining a commitment to technical integrity.High-performance deep spiking neural networks with 0.3 spikes per neuron



SNNs can in principle be applied to the same applications as traditional ANNs.[31] In addition, SNNs can model the central nervous system of biological organisms, such as an insect seeking food without prior knowledge of the environment.[32] Due to their relative realism, they can be used to study the operation of biological neural circuits. Starting with a hypothesis about the topology of a biological neuronal circuit and its function, recordings of this circuit can be compared to the output of the corresponding SNN, evaluating the plausibility of the hypothesis. However, there is a lack of effective training mechanisms for SNNs, which can be inhibitory for some applications, including computer vision tasks.
As of 2019 SNNs lag behind ANNs in terms of accuracy, but the gap is decreasing, and has vanished on some tasks.[33]
When using SNNs for image based data, the images need to be converted into binary spike trains.[34] Types of encodings include:[35]

Temporal coding; generating one spike per neuron, in which spike latency is inversely proportional to the pixel intensity.
Rate coding: converting pixel intensity into a spike train, where the number of spikes is proportional to the pixel intensity.
Direct coding; using a trainable layer to generate a floating-point value for each time step. The layer converts each pixel at a certain time step into a floating-point value, and then a threshold is used on the generated floating-point values to pick either zero or one.
Phase coding; encoding temporal information into spike patterns based on a global oscillator.
Burst coding; transmitting a burst of spikes in a small time, increasing the reliability of synaptic communication between neurons.
However, Spiking Neural Networks are very sensitive to their parameters, like the membrane's threshold, decay rate, or  slope of the sigmoid function that approximates the Heavyside step function to be differentiable. Some works performed optimizations to overcome the network's sensibility.[36]



A diverse range of application software can simulate SNNs. This software can be classified according to its uses:


Unsupervised learning with ferroelectric synapses These simulate complex neural models with a high level of detail and accuracy. Large networks usually require lengthy processing. Candidates include:[37]
Brian – developed by Romain Brette and Dan Goodman at the École Normale Supérieure;
GENESIS (the GEneral NEural SImulation System[38]) – developed in James Bower's laboratory at Caltech;
NEST – developed by the NEST Initiative;
NEURON – mainly developed by Michael Hines, John W. Moore and Ted Carnevale in Yale University and Duke University;
RAVSim (Runtime Tool) [39] – mainly developed by Sanaullah in Bielefeld University of Applied Sciences and Arts;

Predicting STDP learning with ferroelectric synapses
Neuron-to-neuron mesh routing model

Future neuromorphic architectures[40] will comprise billions of such nanosynapses, which require a clear understanding of the physical mechanisms responsible for plasticity. Experimental systems based on ferroelectric tunnel junctions have been used to show that STDP can be harnessed from heterogeneous polarization switching. Through combined scanning probe imaging, electrical transport and atomic-scale molecular dynamics, conductance variations can be modelled by nucleation-dominated reversal of domains. Simulations show that arrays of ferroelectric nanosynapses can autonomously learn to recognize patterns in a predictable way, opening the path towards unsupervised learning.[41]

Unsupervised learning with ferroelectric synapses
Akida is a completely digital event-based neural processing device with 1.2 million artificial neurons and 10 billion artificial synapses developed by BrainChip. Utilizing event-based possessing, it analyzes essential inputs at specific points. Results are stored in the on-chip memory units.
Neurogrid is a board that can simulate spiking neural networks directly in hardware. (Stanford University)
SpiNNaker (Spiking Neural Network Architecture) uses ARM processors as the building blocks of a massively parallel computing platform based on a six-layer thalamocortical model. (University of Manchester)[42] The SpiNNaker system is based on numerical models running in real time on custom digital multicore chips using the ARM architecture. It provides custom digital chips, each with eighteen cores and a shared local 128 Mbyte RAM, with a total of over 1,000,000 cores.[43] A single chip can simulate 16,000 neurons with eight million plastic synapses running in real time.[44]
TrueNorth is a processor that contains 5.4 billion transistors that consumes only 70 milliwatts; most processors in personal computers contain about 1.4 billion transistors and require 35 watts or more. IBM refers to the design principle behind TrueNorth as neuromorphic computing. Its primary purpose is pattern recognition. While critics say the chip isn't powerful enough, its supporters point out that this is only the first generation, and the capabilities of improved iterations will become clear. (IBM)[45]

Classification capabilities of spiking networks trained according to unsupervised learning methods[46] have been tested on the common benchmark datasets, such as, Iris, Wisconsin Breast Cancer or Statlog Landsat dataset.[47][48] Various approaches to information encoding and network design have been used. For example, a 2-layer feedforward network for data clustering and classification. Based on the idea proposed in Hopfield (1995) the authors implemented models of local receptive fields combining the properties of radial basis functions (RBF) and spiking neurons to convert input signals (classified data) having a floating-point representation into a spiking representation.[49][50]



CoDi
Cognitive architecture
Cognitive map
Cognitive computer
Computational neuroscience
Neural coding
Neural correlate
Neural decoding
Neuroethology
Neuroinformatics
Models of neural computation
Motion perception
Systems neuroscience



^ a b Maass W (1997). "Networks of spiking neurons: The third generation of neural network models". Neural Networks. 10 (9): 1659–1671. doi:10.1016/S0893-6080(97)00011-7. ISSN 0893-6080.

^ Auge, Daniel; Hille, Julian; Mueller, Etienne; Knoll, Alois (2021-12-01). "A Survey of Encoding Techniques for Signal Processing in Spiking Neural Networks". Neural Processing Letters. 53 (6): 4693–4710. doi:10.1007/s11063-021-10562-2. ISSN 1573-773X.

^ Gerstner W, Kistler WM (2002). Spiking neuron models : single neurons, populations, plasticity. Cambridge, U.K.: Cambridge University Press. ISBN 0-511-07817-X. OCLC 57417395.

^ Wang, Xiangwen; Lin, Xianghong; Dang, Xiaochao (2020-05-01). "Supervised learning in spiking neural networks: A review of algorithms and evaluations". Neural Networks. 125: 258–280. doi:10.1016/j.neunet.2020.02.011. ISSN 0893-6080. PMID 32146356. S2CID 212638634.

^ Taherkhani, Aboozar; Belatreche, Ammar; Li, Yuhua; Cosma, Georgina; Maguire, Liam P.; McGinnity, T. M. (2020-02-01). "A review of learning in biologically plausible spiking neural networks". Neural Networks. 122: 253–272. doi:10.1016/j.neunet.2019.09.036. ISSN 0893-6080. PMID 31726331. S2CID 207904985.

^ a b Ganguly, Chittotosh; Bezugam, Sai Sukruth; Abs, Elisabeth; Payvand, Melika; Dey, Sounak; Suri, Manan (2024-02-01). "Spike frequency adaptation: bridging neural models and neuromorphic applications". Communications Engineering. 3 (1): 22. doi:10.1038/s44172-024-00165-9. ISSN 2731-3395. PMC 11053160.

^ Lee D, Lee G, Kwon D, Lee S, Kim Y, Kim J (June 2018). "Flexon: A Flexible Digital Neuron for Efficient Spiking Neural Network Simulations". 2018 ACM/IEEE 45th Annual International Symposium on Computer Architecture (ISCA). pp. 275–288. doi:10.1109/isca.2018.00032. ISBN 978-1-5386-5984-7. S2CID 50778421.

^ Goodman, D. F., & Brette, R. (2008). Brian: a simulator for spiking neural networks in python. Frontiers in neuroinformatics, 2, 350.

^ Vreeken, J. (2003). Spiking neural networks, an introduction

^ Yamazaki, K.; Vo-Ho, V. K.; Bulsara, D.; Le, N (30 June 2022). "Spiking neural networks and their applications: A review". Brain Sciences. 12 (7): 863. doi:10.3390/brainsci12070863. PMC 9313413. PMID 35884670.

^ Ballard, D. H. (1987, July). Modular learning in neural networks. In Proceedings of the sixth National conference on Artificial intelligence-Volume 1 (pp. 279-284).

^ Peretto, P. (1984). Collective properties of neural networks: a statistical physics approach. Biological cybernetics, 50(1), 51-62.

^ Kurogi, S. (1987). A model of neural network for spatiotemporal pattern recognition. Biological cybernetics, 57(1), 103-114.

^ Anderson, J. A. (1972). A simple neural network generating an interactive memory. Mathematical biosciences, 14(3-4), 197-220.

^ Gerstner W (2001). "Spiking Neurons". In Maass W, Bishop CM (eds.). Pulsed Neural Networks. MIT Press. ISBN 978-0-262-63221-8.

^ Hodgkin, A. L.; Huxley, A. F. (1952-08-28). "A quantitative description of membrane current and its application to conduction and excitation in nerve". The Journal of Physiology. 117 (4): 500–544. doi:10.1113/jphysiol.1952.sp004764. ISSN 0022-3751. PMC 1392413. PMID 12991237.

^ Dan, Yang; Poo, Mu-Ming (July 2006). "Spike Timing-Dependent Plasticity: From Synapse to Perception". Physiological Reviews. 86 (3): 1033–1048. doi:10.1152/physrev.00030.2005. ISSN 0031-9333. PMID 16816145.

^ Nagornov, Nikolay N.; Lyakhov, Pavel A.; Bergerman, Maxim V.; Kalita, Diana I. (2024). "Modern Trends in Improving the Technical Characteristics of Devices and Systems for Digital Image Processing". IEEE Access. 12: 44659–44681. Bibcode:2024IEEEA..1244659N. doi:10.1109/ACCESS.2024.3381493. ISSN 2169-3536.

^ Van Wezel M (2020). A robust modular spiking neural networks training methodology for time-series datasets: With a focus on gesture control (Master of Science thesis). Delft University of Technology.

^ a b c Maass W (1997). "Networks of spiking neurons: The third generation of neural network models". Neural Networks. 10 (9): 1659–1671. doi:10.1016/S0893-6080(97)00011-7.

^ Furber, Steve (August 2016). "Large-scale neuromorphic computing systems". Journal of Neural Engineering. 13 (5): 051001. Bibcode:2016JNEng..13e1001F. doi:10.1088/1741-2560/13/5/051001. ISSN 1741-2552. PMID 27529195.

^ Neftci, Emre O.; Mostafa, Hesham; Zenke, Friedemann (2019). "Surrogate Gradient Learning in Spiking Neural Networks: Bringing the Power of Gradient-Based Optimization to Spiking Neural Networks". IEEE Signal Processing Magazine. 36 (6): 51–63. Bibcode:2019ISPM...36f..51N. doi:10.1109/msp.2019.2931595.

^ Salaj, Darjan; Subramoney, Anand; Kraisnikovic, Ceca; Bellec, Guillaume; Legenstein, Robert; Maass, Wolfgang (2021-07-26). O'Leary, Timothy; Behrens, Timothy E; Gutierrez, Gabrielle (eds.). "Spike frequency adaptation supports network computations on temporally dispersed information". eLife. 10: e65459. doi:10.7554/eLife.65459. ISSN 2050-084X. PMC 8313230. PMID 34310281.

^ Izhikevich, E.M. (2004). "Which model to use for cortical spiking neurons?". IEEE Transactions on Neural Networks. 15 (5): 1063–1070. doi:10.1109/tnn.2004.832719. PMID 15484883. S2CID 7354646. Retrieved 2024-02-14.

^ Adibi, M., McDonald, J. S., Clifford, C. W. & Arabzadeh, E. Adaptation improves neural coding efficiency despite increasing correlations in variability. J. Neurosci. 33, 2108–2120 (2013)

^ Laughlin, S. (1981). "A simple coding procedure enhances a neuron's information capacity". Zeitschrift für Naturforschung C. 36 (9–10): 910–912. ISSN 0341-0382. PMID 7303823.

^ Querlioz, Damien; Bichler, Olivier; Dollfus, Philippe; Gamrat, Christian (2013). "Immunity to Device Variations in a Spiking Neural Network With Memristive Nanodevices". IEEE Transactions on Nanotechnology. 12 (3): 288–295. Bibcode:2013ITNan..12..288Q. doi:10.1109/TNANO.2013.2250995. S2CID 14416573. Retrieved 2024-02-14.

^ Yamazaki, Kashu; Vo-Ho, Viet-Khoa; Bulsara, Darshan; Le, Ngan (July 2022). "Spiking Neural Networks and Their Applications: A Review". Brain Sciences. 12 (7): 863. doi:10.3390/brainsci12070863. ISSN 2076-3425. PMC 9313413. PMID 35884670.

^ Shaban, Ahmed; Bezugam, Sai Sukruth; Suri, Manan (2021-07-09). "An adaptive threshold neuron for recurrent spiking neural networks with nanodevice hardware implementation". Nature Communications. 12 (1): 4234. Bibcode:2021NatCo..12.4234S. doi:10.1038/s41467-021-24427-8. ISSN 2041-1723. PMC 8270926. PMID 34244491.

^ Bellec, Guillaume; Salaj, Darjan; Subramoney, Anand; Legenstein, Robert; Maass, Wolfgang (2018-12-25), Long short-term memory and learning-to-learn in networks of spiking neurons, arXiv:1803.09574

^ Alnajjar F, Murase K (2008). "A simple Aplysia-like spiking neural network to generate adaptive behavior in autonomous robots". Adaptive Behavior. 14 (5): 306–324. doi:10.1177/1059712308093869. S2CID 16577867.

^ Zhang X, Xu Z, Henriquez C, Ferrari S (Dec 2013). "Spike-based indirect training of a spiking neural network-controlled virtual insect". 52nd IEEE Conference on Decision and Control. pp. 6798–6805. CiteSeerX 10.1.1.671.6351. doi:10.1109/CDC.2013.6760966. ISBN 978-1-4673-5717-3. S2CID 13992150.

^ Tavanaei A, Ghodrati M, Kheradpisheh SR, Masquelier T, Maida A (March 2019). "Deep learning in spiking neural networks". Neural Networks. 111: 47–63. arXiv:1804.08150. doi:10.1016/j.neunet.2018.12.002. PMID 30682710. S2CID 5039751.

^ Yamazaki K, Vo-Ho VK, Bulsara D, Le N (June 2022). "Spiking Neural Networks and Their Applications: A Review". Brain Sciences. 12 (7): 863. doi:10.3390/brainsci12070863. PMC 9313413. PMID 35884670.

^ Kim Y, Park H, Moitra A, Bhattacharjee A, Venkatesha Y, Panda P (2022-01-31). "Rate Coding or Direct Coding: Which One is Better for Accurate, Robust, and Energy-efficient Spiking Neural Networks?". arXiv:2202.03133 [cs.NE].

^ Ribeiro, Bernardete; Antunes, Francisco; Perdigão, Dylan; Silva, Catarina (2024-08-05). "Convolutional Spiking Neural Networks targeting learning and inference in highly imbalanced datasets". Pattern Recognition Letters. doi:10.1016/j.patrec.2024.08.002.

^ Abbott LF, Nelson SB (November 2000). "Synaptic plasticity: taming the beast". Nature Neuroscience. 3 (S11): 1178–1183. doi:10.1038/81453. PMID 11127835. S2CID 2048100.

^ Atiya AF, Parlos AG (May 2000). "New results on recurrent network training: unifying the algorithms and accelerating convergence". IEEE Transactions on Neural Networks. 11 (3): 697–709. doi:10.1109/72.846741. PMID 18249797.

^ Sanaullah S, Koravuna S, Rückert U, Jungeblut T (August 2023). "Evaluation of Spiking Neural Nets-Based Image Classification Using the Runtime Simulator RAVSim". International Journal of Neural Systems. 33 (9): 2350044. doi:10.1142/S0129065723500442. PMID 37604777. S2CID 259445644.

^ Sutton RS, Barto AG (2002) Reinforcement Learning: An Introduction. Bradford Books, MIT Press, Cambridge, MA.

^ Boyn S, Grollier J, Lecerf G, Xu B, Locatelli N, Fusil S, et al. (April 2017). "Learning through ferroelectric domain dynamics in solid-state synapses". Nature Communications. 8: 14736. Bibcode:2017NatCo...814736B. doi:10.1038/ncomms14736. PMC 5382254. PMID 28368007.

^ Jin X, Furber SB, Woods JV (2008). "Efficient modelling of spiking neural networks on a scalable chip multiprocessor". 2008 IEEE International Joint Conference on Neural Networks (IEEE World Congress on Computational Intelligence). pp. 2812–2819. doi:10.1109/IJCNN.2008.4634194. ISBN 978-1-4244-1820-6. S2CID 2103654.

^ "Neuromorphic Computing". Human Brain Project.

^ "Hardware: Available Systems". Human Brain Project. Retrieved 2020-05-10.

^ Markoff J (8 August 2014). "A new chip functions like a brain, IBM says". The New York Times. p. B1.

^ Ponulak F, Kasiński A (February 2010). "Supervised learning in spiking neural networks with ReSuMe: sequence learning, classification, and spike shifting". Neural Computation. 22 (2): 467–510. doi:10.1162/neco.2009.11-08-901. PMID 19842989. S2CID 12572538.

^ Newman D, Hettich S, Blake C, Merz C (1998). "UCI repository of machine learning databases".

^ Bohte S, Kok JN, La Poutré H (2002). "Error-backpropagation in temporally encoded networks of spiking neurons". Neurocomputing. 48 (1–4): 17–37. doi:10.1016/S0925-2312(01)00658-0.

^ Pfister JP, Toyoizumi T, Barber D, Gerstner W (June 2006). "Optimal spike-timing-dependent plasticity for precise action potential firing in supervised learning". Neural Computation. 18 (6): 1318–1348. arXiv:q-bio/0502037. Bibcode:2005q.bio.....2037P. doi:10.1162/neco.2006.18.6.1318. PMID 16764506. S2CID 6379045.

^ Bohte SM, La Poutré H, Kok JN (March 2002). "Unsupervised clustering with spiking neurons by sparse temporal coding and multilayer RBF networks". IEEE Transactions on Neural Networks. 13 (2): 426–435. doi:10.1109/72.991428. PMID 18244443.






