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TopChaotic Neural Networks And Incremental Learning
The incremental learning was developed by using the chaotic neurons. The chaotic neurons and the chaotic neural networks were proposed by Aihara (Aihara, Tanabe &Toyoda, 1990).
We presented the incremental learning that provides an associative memory (Asakawa, Deguchi & Ishii, 2001; Deguchi & Ishii, 2004; Deguchi, Fukuta & Ishii, 2013; Deguchi, Matsuo, Kimura & Ishii, 2009-07; Deguchi, Takahashi & Ishii, 2014). The network type is an interconnected network, in which each neuron receives one external input, and is defined as follows (Aihara, Tanabe & Toyoda, 1990):
,
(1),
(2),
(3),
(4) where
![ijsi.2014100106.m05](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m05.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
is the output of the
![ijsi.2014100106.m06](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m06.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
-th neuron at time
![ijsi.2014100106.m07](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m07.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
,
![ijsi.2014100106.m08](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m08.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
is the output sigmoid function described below in (5),
![ijsi.2014100106.m09](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m09.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
,
![ijsi.2014100106.m10](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m10.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
,
![ijsi.2014100106.m11](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m11.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
are the time decay constants,
![ijsi.2014100106.m12](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m12.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
is the input to the
![ijsi.2014100106.m13](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m13.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
-th neuron at time
![ijsi.2014100106.m14](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m14.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
,
![ijsi.2014100106.m15](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m15.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
is the weight for external inputs,
![ijsi.2014100106.m16](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m16.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
is the size—the number of the neurons in the network,
![ijsi.2014100106.m17](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m17.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
is the connection weight from the
![ijsi.2014100106.m18](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m18.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
-th neuron to the
![ijsi.2014100106.m19](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m19.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
-th neuron, and
![ijsi.2014100106.m20](https://igiprodst.blob.core.windows.net:443/source-content/9781466656628_91701/ijsi.2014100106.m20.png?sv=2015-12-11&sr=c&sig=19Ihrv3wquhYmRGYY%2BLbDPr4uZ%2FixOP%2FwO1nskGisKc%3D&se=2019-11-16T11%3A33%3A13Z&sp=r)
is the parameter that specifies the relation between the neuron output and the refractoriness.