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AlexCrimi<p><a href="https://mstdn.social/tags/simplicialcomplex" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>simplicialcomplex</span></a> + <a href="https://mstdn.social/tags/Causality" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Causality</span></a> +<a href="https://mstdn.social/tags/Reservoircomputing" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Reservoircomputing</span></a>:<br>"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" translate="no" target="_blank"><span class="invisible">https://www.</span><span class="ellipsis">nature.com/articles/s41467-024</span><span class="invisible">-46852-1</span></a></p><p><a href="https://mstdn.social/tags/dynamicalsystem" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>dynamicalsystem</span></a> <a href="https://mstdn.social/tags/ML" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>ML</span></a> <a href="https://mstdn.social/tags/AI" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>AI</span></a></p>
Brain Dynamics Toolbox<p>Saddles and stable nodes in a nonlinear <a href="https://mastodon.au/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>DynamicalSystem</span></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" target="_blank">#<span>NonlinearDynamics</span></a> and <a href="https://mastodon.au/tags/Chaos" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Chaos</span></a>. Simulated with <a href="https://mastodon.au/tags/BrainDynamicsToolbox" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>BrainDynamicsToolbox</span></a>.</p>
Brain Dynamics Toolbox<p>The damped and driven <a href="https://mastodon.au/tags/pendulum" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>pendulum</span></a> is a classic <a href="https://mastodon.au/tags/nonlinear" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>nonlinear</span></a> <a href="https://mastodon.au/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>DynamicalSystem</span></a> in physics. Simulated here with the <a href="https://mastodon.au/tags/BrainDynamicsToolbox" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>BrainDynamicsToolbox</span></a>.</p>
Brain Dynamics Toolbox<p>The damped and driven <a href="https://mastodon.au/tags/pendulum" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>pendulum</span></a> is a classic <a href="https://mastodon.au/tags/nonlinear" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>nonlinear</span></a> <a href="https://mastodon.au/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>DynamicalSystem</span></a> in physics. Simulated here with the <a href="https://mastodon.au/tags/BrainDynamicsToolbox" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>BrainDynamicsToolbox</span></a>.</p>
Brain Dynamics Toolbox<p>Saddles and stable nodes in a nonlinear <a href="https://mastodon.au/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>DynamicalSystem</span></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" target="_blank">#<span>NonlinearDynamics</span></a> and <a href="https://mastodon.au/tags/Chaos" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Chaos</span></a>. Simulated with <a href="https://mastodon.au/tags/BrainDynamicsToolbox" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>BrainDynamicsToolbox</span></a>.</p>
Brain Dynamics Toolbox<p>The <a href="https://mastodon.au/tags/Hopf" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Hopf</span></a> bifurcation describes the birth of a <a href="https://mastodon.au/tags/LimitCycle" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>LimitCycle</span></a> in a <a href="https://mastodon.au/tags/DynamicalSystem" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>DynamicalSystem</span></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" target="_blank">#<span>BrainDynamicsToolbox</span></a>.</p>
Fabrizio Musacchio<p>An important step in <a href="https://sigmoid.social/tags/ComputationalNeuroscience" class="mention hashtag" rel="tag">#<span>ComputationalNeuroscience</span></a> 🧠💻 was the development of the <a href="https://sigmoid.social/tags/HodgkinHuxley" class="mention hashtag" rel="tag">#<span>HodgkinHuxley</span></a> model, for which Hodgkin and Huxley received the <a href="https://sigmoid.social/tags/NobelPrize" class="mention hashtag" rel="tag">#<span>NobelPrize</span></a> in 1963. The model describes the dynamics of the <a href="https://sigmoid.social/tags/MembranePotential" class="mention hashtag" rel="tag">#<span>MembranePotential</span></a> of a <a href="https://sigmoid.social/tags/neuron" class="mention hashtag" rel="tag">#<span>neuron</span></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="tag">#<span>Python</span></a>: </p><p>🌍 <a href="https://www.fabriziomusacchio.com/blog/2024-04-21-hodgkin_huxley_model/" target="_blank" rel="nofollow noopener" translate="no"><span class="invisible">https://www.</span><span class="ellipsis">fabriziomusacchio.com/blog/202</span><span class="invisible">4-04-21-hodgkin_huxley_model/</span></a></p><p>Feel free to share and to experiment with the code.</p><p><a href="https://sigmoid.social/tags/CompNeuro" class="mention hashtag" rel="tag">#<span>CompNeuro</span></a> <a href="https://sigmoid.social/tags/PythonTutorial" class="mention hashtag" rel="tag">#<span>PythonTutorial</span></a> <a href="https://sigmoid.social/tags/NeuralDynamics" class="mention hashtag" rel="tag">#<span>NeuralDynamics</span></a> <a href="https://sigmoid.social/tags/DynamicalSystem" class="mention hashtag" rel="tag">#<span>DynamicalSystem</span></a></p>
Fabrizio Musacchio<p>Here is another <a href="https://sigmoid.social/tags/PhasePlaneAnalysis" class="mention hashtag" rel="tag">#<span>PhasePlaneAnalysis</span></a> <a href="https://sigmoid.social/tags/tutorial" class="mention hashtag" rel="tag">#<span>tutorial</span></a>, this time applied to the <a href="https://sigmoid.social/tags/VanDerPolOscillator" class="mention hashtag" rel="tag">#<span>VanDerPolOscillator</span></a>, a non-conservative <a href="https://sigmoid.social/tags/oscillator" class="mention hashtag" rel="tag">#<span>oscillator</span></a> with nonlinear damping:</p><p>🌍 <a href="https://www.fabriziomusacchio.com/blog/2024-03-24-van_der_pol_oscillator/" target="_blank" rel="nofollow noopener" translate="no"><span class="invisible">https://www.</span><span class="ellipsis">fabriziomusacchio.com/blog/202</span><span class="invisible">4-03-24-van_der_pol_oscillator/</span></a></p><p><a href="https://sigmoid.social/tags/DynamicalSystem" class="mention hashtag" rel="tag">#<span>DynamicalSystem</span></a> <a href="https://sigmoid.social/tags/ComputationalScience" class="mention hashtag" rel="tag">#<span>ComputationalScience</span></a> <a href="https://sigmoid.social/tags/PhasePortraits" class="mention hashtag" rel="tag">#<span>PhasePortraits</span></a> <a href="https://sigmoid.social/tags/Python" class="mention hashtag" rel="tag">#<span>Python</span></a></p>
Fabrizio Musacchio<p>Exploring the behavior of <a href="https://sigmoid.social/tags/DynamicalSystems" class="mention hashtag" rel="tag">#<span>DynamicalSystems</span></a> directly through their differential equations can be complex. <a href="https://sigmoid.social/tags/PhasePlaneAnalysis" class="mention hashtag" rel="tag">#<span>PhasePlaneAnalysis</span></a> offers a clearer and intuitive view by visualizing dynamics with <a href="https://sigmoid.social/tags/PhasePortraits" class="mention hashtag" rel="tag">#<span>PhasePortraits</span></a>, simplifying understanding. Here is a <a href="https://sigmoid.social/tags/tutorial" class="mention hashtag" rel="tag">#<span>tutorial</span></a> along with some <a href="https://sigmoid.social/tags/Python" class="mention hashtag" rel="tag">#<span>Python</span></a> code, exploring this method and exemplarily applying it to the simple pendulum.</p><p>🌍 <a href="https://www.fabriziomusacchio.com/blog/2024-03-17-phase_plane_analysis/" target="_blank" rel="nofollow noopener" translate="no"><span class="invisible">https://www.</span><span class="ellipsis">fabriziomusacchio.com/blog/202</span><span class="invisible">4-03-17-phase_plane_analysis/</span></a></p><p><a href="https://sigmoid.social/tags/ChaoticSystems" class="mention hashtag" rel="tag">#<span>ChaoticSystems</span></a> <a href="https://sigmoid.social/tags/DynamicalSystem" class="mention hashtag" rel="tag">#<span>DynamicalSystem</span></a> <a href="https://sigmoid.social/tags/ComputationalScience" class="mention hashtag" rel="tag">#<span>ComputationalScience</span></a></p>
Non-Euclidean Dreamer<p>Genuary 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 )</p><p>Full-Res full-length full-size version: <a href="https://youtu.be/q8V0KPQRjRM" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="">youtu.be/q8V0KPQRjRM</span><span class="invisible"></span></a></p><p><a href="https://mathstodon.xyz/tags/genuary" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>genuary</span></a> <a href="https://mathstodon.xyz/tags/genuary5" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>genuary5</span></a> <a href="https://mathstodon.xyz/tags/genuary2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>genuary2024</span></a> <a href="https://mathstodon.xyz/tags/dynamicalsystem" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>dynamicalsystem</span></a></p>
Non-Euclidean Dreamer<p>Proudly presenting this month's High-res Render for Patrons of Level Square and up: full size 25600×25600 pixels</p><p>If you try to "magic eye" it, it shimmers!</p><p>It's my favorite of the iteration functions I explored: (x,y)=(x-t*tan(y),x).<br>The squares have a side length of pi, due to tan of course.<br>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.</p><p><a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathart</span></a> <a href="https://mathstodon.xyz/tags/codeart" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>codeart</span></a> <a href="https://mathstodon.xyz/tags/fractal" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>fractal</span></a> <a href="https://mathstodon.xyz/tags/blackandwhite" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>blackandwhite</span></a> <a href="https://mathstodon.xyz/tags/dynamicalsystem" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>dynamicalsystem</span></a> <a href="https://mathstodon.xyz/tags/mastoart" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mastoart</span></a></p>
Non-Euclidean Dreamer<p>Bogdanov Map, I am so glad I found you!</p><p><a href="https://mathstodon.xyz/tags/mastoArt" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mastoArt</span></a> <a href="https://mathstodon.xyz/tags/codeArt" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>codeArt</span></a> <a href="https://mathstodon.xyz/tags/dynamicalsystem" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>dynamicalsystem</span></a> <a href="https://mathstodon.xyz/tags/attractor" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>attractor</span></a> <a href="https://mathstodon.xyz/tags/bogdanov" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>bogdanov</span></a></p>