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AlexCrimi<a href="https://mstdn.social/tags/simplicialcomplex" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#simplicialcomplex</a> + <a href="https://mstdn.social/tags/Causality" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Causality</a> +<a href="https://mstdn.social/tags/Reservoircomputing" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Reservoircomputing</a>: "Higher-order Granger reservoir computing: simultaneously achieving scalable complex structures inference and accurate dynamics prediction" <a href="https://www.nature.com/articles/s41467-024-46852-1" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://www.nature.com/articles/s41467-024-46852-1</a><a href="https://mstdn.social/tags/dynamicalsystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#dynamicalsystem</a> <a href="https://mstdn.social/tags/ML" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ML</a> <a href="https://mstdn.social/tags/AI" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AI</a>

Brain Dynamics ToolboxSaddles and stable nodes in a nonlinear <a href="https://mastodon.au/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystem</a>. From example 6.3.1 in Steven Strogatz's textbook on <a href="https://mastodon.au/tags/NonlinearDynamics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#NonlinearDynamics</a> and <a href="https://mastodon.au/tags/Chaos" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Chaos</a>. Simulated with <a href="https://mastodon.au/tags/BrainDynamicsToolbox" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BrainDynamicsToolbox</a>.

Brain Dynamics ToolboxThe damped and driven <a href="https://mastodon.au/tags/pendulum" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#pendulum</a> is a classic <a href="https://mastodon.au/tags/nonlinear" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#nonlinear</a> <a href="https://mastodon.au/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystem</a> in physics. Simulated here with the <a href="https://mastodon.au/tags/BrainDynamicsToolbox" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BrainDynamicsToolbox</a>.

Brain Dynamics ToolboxThe <a href="https://mastodon.au/tags/Hopf" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Hopf</a> bifurcation describes the birth of a <a href="https://mastodon.au/tags/LimitCycle" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#LimitCycle</a> in a <a href="https://mastodon.au/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystem</a>. Here is the normal form in euclidean coordinates, simulated with the <a href="https://mastodon.au/tags/BrainDynamicsToolbox" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BrainDynamicsToolbox</a>.

Fabrizio MusacchioAn important step in <a href="https://sigmoid.social/tags/ComputationalNeuroscience" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ComputationalNeuroscience</a> 🧠💻 was the development of the <a href="https://sigmoid.social/tags/HodgkinHuxley" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#HodgkinHuxley</a> model, for which Hodgkin and Huxley received the <a href="https://sigmoid.social/tags/NobelPrize" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#NobelPrize</a> in 1963. The model describes the dynamics of the <a href="https://sigmoid.social/tags/MembranePotential" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#MembranePotential</a> of a <a href="https://sigmoid.social/tags/neuron" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#neuron</a> 🔬 by incorporating biophysiological properties. See here how it is derived, along with a simple implementation in <a href="https://sigmoid.social/tags/Python" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Python</a>: 🌍 <a href="https://www.fabriziomusacchio.com/blog/2024-04-21-hodgkin_huxley_model/" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://www.fabriziomusacchio.com/blog/2024-04-21-hodgkin_huxley_model/</a>Feel free to share and to experiment with the code.<a href="https://sigmoid.social/tags/CompNeuro" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#CompNeuro</a> <a href="https://sigmoid.social/tags/PythonTutorial" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PythonTutorial</a> <a href="https://sigmoid.social/tags/NeuralDynamics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#NeuralDynamics</a> <a href="https://sigmoid.social/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystem</a>

Fabrizio MusacchioHere is another <a href="https://sigmoid.social/tags/PhasePlaneAnalysis" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PhasePlaneAnalysis</a> <a href="https://sigmoid.social/tags/tutorial" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#tutorial</a>, this time applied to the <a href="https://sigmoid.social/tags/VanDerPolOscillator" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#VanDerPolOscillator</a>, a non-conservative <a href="https://sigmoid.social/tags/oscillator" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#oscillator</a> with nonlinear damping:🌍 <a href="https://www.fabriziomusacchio.com/blog/2024-03-24-van_der_pol_oscillator/" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://www.fabriziomusacchio.com/blog/2024-03-24-van_der_pol_oscillator/</a><a href="https://sigmoid.social/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystem</a> <a href="https://sigmoid.social/tags/ComputationalScience" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ComputationalScience</a> <a href="https://sigmoid.social/tags/PhasePortraits" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PhasePortraits</a> <a href="https://sigmoid.social/tags/Python" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Python</a>

Brain Dynamics Toolbox<a href="https://mastodon.au/tags/PhasePlane" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PhasePlane</a> of a symmetrical <a href="https://mastodon.au/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystem</a> with a <a href="https://mastodon.au/tags/homoclinic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#homoclinic</a> orbit. From example 6.6.2 of Steven Strogatz's textbook on <a href="https://mastodon.au/tags/NonlinearDynamics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#NonlinearDynamics</a> and Chaos. Simulated with the <a href="https://mastodon.au/tags/BrainDynamicsToolbox" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BrainDynamicsToolbox</a>.

Fabrizio MusacchioExploring the behavior of <a href="https://sigmoid.social/tags/DynamicalSystems" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystems</a> directly through their differential equations can be complex. <a href="https://sigmoid.social/tags/PhasePlaneAnalysis" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PhasePlaneAnalysis</a> offers a clearer and intuitive view by visualizing dynamics with <a href="https://sigmoid.social/tags/PhasePortraits" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PhasePortraits</a>, simplifying understanding. Here is a <a href="https://sigmoid.social/tags/tutorial" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#tutorial</a> along with some <a href="https://sigmoid.social/tags/Python" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Python</a> code, exploring this method and exemplarily applying it to the simple pendulum.🌍 <a href="https://www.fabriziomusacchio.com/blog/2024-03-17-phase_plane_analysis/" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://www.fabriziomusacchio.com/blog/2024-03-17-phase_plane_analysis/</a><a href="https://sigmoid.social/tags/ChaoticSystems" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ChaoticSystems</a> <a href="https://sigmoid.social/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystem</a> <a href="https://sigmoid.social/tags/ComputationalScience" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ComputationalScience</a>

Brain Dynamics ToolboxThe <a href="https://mastodon.au/tags/stability" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#stability</a> of a <a href="https://mastodon.au/tags/FixedPoint" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#FixedPoint</a> in a <a href="https://mastodon.au/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystem</a> is determined by the <a href="https://mastodon.au/tags/Trace" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Trace</a> and <a href="https://mastodon.au/tags/Determinant" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Determinant</a> of its <a href="https://mastodon.au/tags/Jacobian" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Jacobian</a> matrix. This is example 5.2.6 from Steven Strogatz's textbook on <a href="https://mastodon.au/tags/NonlinearDynamics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#NonlinearDynamics</a> and Chaos. Simulated with <a href="https://mastodon.au/tags/BrainDynamicsToolbox" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BrainDynamicsToolbox</a>.

Non-Euclidean DreamerGenuary Prompt Nr. 5 is "In the style of Vera Molnàr". When I looked at her works I liked the framing squares with things going on in them. They reminded me at what I saw when investigating Dynamical Systems. This is 8 iterations of the function f(x,y)=( x-(1+y/4)tan(y)-t*y , x )Full-Res full-length full-size version: <a href="https://youtu.be/q8V0KPQRjRM" target="_blank" rel="nofollow noopener noreferrer" translate="no">https://youtu.be/q8V0KPQRjRM</a><a href="https://mathstodon.xyz/tags/genuary" class="mention hashtag" rel="tag">#genuary</a> <a href="https://mathstodon.xyz/tags/genuary5" class="mention hashtag" rel="tag">#genuary5</a> <a href="https://mathstodon.xyz/tags/genuary2024" class="mention hashtag" rel="tag">#genuary2024</a> <a href="https://mathstodon.xyz/tags/dynamicalsystem" class="mention hashtag" rel="tag">#dynamicalsystem</a>

kandidMapping the complex plane. Using an image of a flower for coloring.<a href="https://gitlab.com/metagrowing/ana/-/blob/master/visual_server/media/frag/cmplx/cmplx-03.frag" rel="nofollow noopener noreferrer" target="_blank">https://gitlab.com/metagrowing/ana/-/blob/master/visual_server/media/frag/cmplx/cmplx-03.frag</a><a href="https://gitlab.com/metagrowing/ana/-/blob/master/live_coding/src/demo/cmplx/cmplx-03.clj" rel="nofollow noopener noreferrer" target="_blank">https://gitlab.com/metagrowing/ana/-/blob/master/live_coding/src/demo/cmplx/cmplx-03.clj</a><a href="https://chaos.social/tags/dynamicalsystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#dynamicalsystem</a> <a href="https://chaos.social/tags/openFrameworks" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#openFrameworks</a>

kandid<a href="https://mathstodon.xyz/@noneuclideandreamer" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@noneuclideandreamer</a> Again, this iterative mapping in the complex plane. This time with adjustable color spectrum and denoised.for(int l=0; l<9; ++l) { _xy = vec2(_xy.y + sin(t * _xy.x), _xy.x); }<a href="https://chaos.social/tags/dynamicalsystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#dynamicalsystem</a>

Non-Euclidean DreamerSoapy Function:(x,y)=(y+tlog|x|,x)(better resolution on youtube)<a href="https://mathstodon.xyz/tags/mastoart" class="mention hashtag" rel="tag">#mastoart</a> <a href="https://mathstodon.xyz/tags/codeart" class="mention hashtag" rel="tag">#codeart</a> <a href="https://mathstodon.xyz/tags/dynamicalsystem" class="mention hashtag" rel="tag">#dynamicalsystem</a>

Non-Euclidean DreamerProudly presenting this month's High-res Render for Patrons of Level Square and up: full size 25600×25600 pixelsIf you try to "magic eye" it, it shimmers!It's my favorite of the iteration functions I explored: (x,y)=(x-t*tan(y),x). The squares have a side length of pi, due to tan of course. Squares below each other would actually look the same since it does only depend on y%pi via tan. So I discretely jumped my t-value from 1.1 over 1 to 0.9.<a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#mathart</a> <a href="https://mathstodon.xyz/tags/codeart" class="mention hashtag" rel="tag">#codeart</a> <a href="https://mathstodon.xyz/tags/fractal" class="mention hashtag" rel="tag">#fractal</a> <a href="https://mathstodon.xyz/tags/blackandwhite" class="mention hashtag" rel="tag">#blackandwhite</a> <a href="https://mathstodon.xyz/tags/dynamicalsystem" class="mention hashtag" rel="tag">#dynamicalsystem</a> <a href="https://mathstodon.xyz/tags/mastoart" class="mention hashtag" rel="tag">#mastoart</a>

Non-Euclidean DreamerThis palette took me by surprise. 😅Movie not compressable to masto size, so stills it is!<a href="https://mathstodon.xyz/tags/mastoart" class="mention hashtag" rel="tag">#mastoart</a> <a href="https://mathstodon.xyz/tags/codeart" class="mention hashtag" rel="tag">#codeart</a> <a href="https://mathstodon.xyz/tags/dynamicalsystem" class="mention hashtag" rel="tag">#dynamicalsystem</a> <a href="https://mathstodon.xyz/tags/attractor" class="mention hashtag" rel="tag">#attractor</a> <a href="https://mathstodon.xyz/tags/fractal" class="mention hashtag" rel="tag">#fractal</a> (mostly) <a href="https://mathstodon.xyz/tags/blackandwhite" class="mention hashtag" rel="tag">#blackandwhite</a>

Non-Euclidean DreamerBogdanov Map, I am so glad I found you!<a href="https://mathstodon.xyz/tags/mastoArt" class="mention hashtag" rel="tag">#mastoArt</a> <a href="https://mathstodon.xyz/tags/codeArt" class="mention hashtag" rel="tag">#codeArt</a> <a href="https://mathstodon.xyz/tags/dynamicalsystem" class="mention hashtag" rel="tag">#dynamicalsystem</a> <a href="https://mathstodon.xyz/tags/attractor" class="mention hashtag" rel="tag">#attractor</a> <a href="https://mathstodon.xyz/tags/bogdanov" class="mention hashtag" rel="tag">#bogdanov</a>

ᛕᎥᕼᗷᗴᖇᑎᗴ丅Ꭵᑕᔕ<a href="https://qoto.org/@karlo" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@karlo</a>The <a href="https://qoto.org/tags/state" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#state</a> I’m talking about is the present state of every living <a href="https://qoto.org/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystem</a> which is controlled by the closed autopoietic process of <a href="https://qoto.org/tags/growth" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#growth</a> and <a href="https://qoto.org/tags/learning" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#learning</a>:

ᛕᎥᕼᗷᗴᖇᑎᗴ丅Ꭵᑕᔕ<a href="https://qoto.org/tags/Meaning" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Meaning</a> is usually described with <a href="https://qoto.org/tags/VectorSpace" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#VectorSpace</a> <a href="https://qoto.org/tags/Semantics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Semantics</a> as in the article below comparing the works from <a href="https://qoto.org/tags/CAShannon" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#CAShannon</a> and <a href="https://qoto.org/tags/AMTuring" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AMTuring</a>:<a href="https://www.journals.uchicago.edu/doi/full/10.1093/bjps/axx029" rel="nofollow noopener noreferrer" target="_blank">https://www.journals.uchicago.edu/doi/full/10.1093/bjps/axx029</a>Basically, what vector space semantics says is that the meaning of a message depends on the <a href="https://qoto.org/tags/Context" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Context</a> provided by the sender’s and the receiver’s <a href="https://qoto.org/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicalSystem</a> <a href="https://qoto.org/tags/Knowledge" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Knowledge</a> <a href="https://qoto.org/tags/State" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#State</a>.As they are two different physical entities they will obviously be in different states, so the two meaning can never be exactly the same.

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