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{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# TD6 Biochemical Programming\n",
    "\n",
    "* CRN synthesis from\n",
    " * GPAC circuits\n",
    " * PIVP definitions of function of time\n",
    " * PIVP definition of function of input variable\n",
    "* Trigonometric functions\n",
    "* Sigmoid functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# GPAC generating cosine(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "compile_wgpac(y::integral integral-1*y)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "present(y,1)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fast*[x2]*[y]for _=[x2+y]=>x1.\r\n",
       "fast*[x1]for x1=>_.\r\n",
       "_=[x1]=>x0.\r\n",
       "_=[x0]=>y.\r\n",
       "present(x2,-1).\r\n",
       "present(y,1).\r\n",
       "parameter(\r\n",
       "  fast = 1000\r\n",
       ").\r\n"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list_model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align*}\n",
       "y_0 &= 1\\\\\n",
       "x0_0 &= 0\\\\\n",
       "x1_0 &= 0\\\\\n",
       "x2_0 &= -1\\\\\n",
       "fast &= 1000\\\\\n",
       "\\frac{dy}{dt} &= x0\\\\\n",
       "\\frac{dx0}{dt} &= x1\\\\\n",
       "\\frac{dx1}{dt} &= fast*[x2]*[y]-fast*[x1]\\\\\n",
       "\\frac{dx2}{dt} &= 0\\\\\n",
       "\\end{align*}\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "list_ode."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerical_simulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    var el = document.getElementById(null);\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) { callback() });\n    }\n    finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.info(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(js_urls, callback) {\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.log(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.log(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = js_urls.length;\n    for (var i = 0; i < js_urls.length; i++) {\n      var url = js_urls[i];\n      var s = document.createElement('script');\n      s.src = url;\n      s.async = false;\n      s.onreadystatechange = s.onload = function() {\n        root._bokeh_is_loading--;\n        if (root._bokeh_is_loading === 0) {\n          console.log(\"Bokeh: all BokehJS libraries loaded\");\n          run_callbacks()\n        }\n      };\n      s.onerror = function() {\n        console.warn(\"failed to load library \" + url);\n      };\n      console.log(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.getElementsByTagName(\"head\")[0].appendChild(s);\n    }\n  };\n\n  var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-0.12.14.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.12.14.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-0.12.14.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-0.12.14.min.js\"];\n\n  var inline_js = [\n    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\n    \n    function(Bokeh) {\n      \n    },\n    function(Bokeh) {\n      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-0.12.14.min.css\");\n      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-0.12.14.min.css\");\n      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.12.14.min.css\");\n      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.12.14.min.css\");\n      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-tables-0.12.14.min.css\");\n      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-tables-0.12.14.min.css\");\n    }\n  ];\n\n  function run_inline_js() {\n    \n    if ((root.Bokeh !== undefined) || (force === true)) {\n      for (var i = 0; i < inline_js.length; i++) {\n        inline_js[i].call(root, root.Bokeh);\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    } else if (force !== true) {\n      var cell = $(document.getElementById(null)).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(js_urls, function() {\n      console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
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       "  var render_items = [{\"docid\":\"fd4adaf9-e7f4-4d9d-949b-8daee4c39fdb\",\"elementid\":\"5d2b1a39-96f4-41b9-b7f7-78731445c9fe\",\"modelid\":\"7f0fd7f1-31e7-4291-9088-c1b54e9e82dd\",\"notebook_comms_target\":\"f5dcf923-b4f2-49ee-a9fe-8029de7aeb7f\"}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "7f0fd7f1-31e7-4291-9088-c1b54e9e82dd"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PIVP generation of cosine(time)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "clear_model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "compile_from_expression(cos, time, f)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "_=[z2_p]=>f_p.\r\n",
       "_=[z2_m]=>f_m.\r\n",
       "_=[f_m]=>z2_p.\r\n",
       "_=[f_p]=>z2_m.\r\n",
       "fast*z2_m*z2_p for z2_m+z2_p=>_.\r\n",
       "fast*f_m*f_p for f_m+f_p=>_.\r\n",
       "present(f_p,1).\r\n",
       "parameter(\r\n",
       "  fast = 1000\r\n",
       ").\r\n"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list_model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align*}\n",
       "f_p_0 &= 1\\\\\n",
       "f_m_0 &= 0\\\\\n",
       "z2_p_0 &= 0\\\\\n",
       "z2_m_0 &= 0\\\\\n",
       "fast &= 1000\\\\\n",
       "\\frac{df_p}{dt} &= z2_p-fast*f_m*f_p\\\\\n",
       "\\frac{df_m}{dt} &= z2_m-fast*f_m*f_p\\\\\n",
       "\\frac{dz2_p}{dt} &= f_m-fast*z2_m*z2_p\\\\\n",
       "\\frac{dz2_m}{dt} &= f_p-fast*z2_m*z2_p\\\\\n",
       "\\end{align*}\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "list_ode."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerical_simulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div class=\"bk-root\">\n",
       "    <div class=\"bk-plotdiv\" id=\"990d94ad-310b-43cf-8923-4b727774a275\"></div>\n",
       "</div>"
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     },
     "metadata": {},
     "output_type": "display_data"
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       "(function(root) {\n",
       "  function embed_document(root) {\n",
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       "  var render_items = [{\"docid\":\"d7aade50-759c-49b9-8baa-8e2bb2001678\",\"elementid\":\"990d94ad-310b-43cf-8923-4b727774a275\",\"modelid\":\"c861a2db-8d91-4e08-b121-14851399ee31\",\"notebook_comms_target\":\"b37e14f9-0a63-4d4e-b62c-035c56cc8794\"}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "c861a2db-8d91-4e08-b121-14851399ee31"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerical_simulation(method:ssa)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div class=\"bk-root\">\n",
       "    <div class=\"bk-plotdiv\" id=\"5f741f21-65d7-4a86-be45-5e3cb832b92c\"></div>\n",
       "</div>"
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     "metadata": {},
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      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
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Application\",\"version\":\"0.12.14\"}};\n",
       "  var render_items = [{\"docid\":\"b956cf3f-38e6-4fda-8ae4-9e59831bd4f3\",\"elementid\":\"5f741f21-65d7-4a86-be45-5e3cb832b92c\",\"modelid\":\"99ad2782-d714-4171-8378-718095210f97\",\"notebook_comms_target\":\"934d3a0f-554d-4e61-9655-81f2ee561cff\"}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "99ad2782-d714-4171-8378-718095210f97"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PIVP computation of cosine(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "clear_model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "compile_from_expression(cos, x, y)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "present(x_p,4)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "_=[z2_m+x_m]=>y_p.\r\n",
       "_=[z2_p+x_p]=>y_p.\r\n",
       "_=[z2_m+x_p]=>y_m.\r\n",
       "_=[z2_p+x_m]=>y_m.\r\n",
       "_=[y_p+x_m]=>z2_p.\r\n",
       "_=[y_m+x_p]=>z2_p.\r\n",
       "_=[y_m+x_m]=>z2_m.\r\n",
       "_=[y_p+x_p]=>z2_m.\r\n",
       "_=[x_m]=>x_p.\r\n",
       "_=[x_p]=>x_m.\r\n",
       "fast*x_m*x_p for x_m+x_p=>_.\r\n",
       "fast*z2_m*z2_p for z2_m+z2_p=>_.\r\n",
       "fast*y_m*y_p for y_m+y_p=>_.\r\n",
       "present(y_p,1).\r\n",
       "present(x_p,4).\r\n",
       "parameter(\r\n",
       "  fast = 1000\r\n",
       ").\r\n"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list_model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align*}\n",
       "y_p_0 &= 1\\\\\n",
       "y_m_0 &= 0\\\\\n",
       "z2_p_0 &= 0\\\\\n",
       "z2_m_0 &= 0\\\\\n",
       "x_p_0 &= 4\\\\\n",
       "x_m_0 &= 0\\\\\n",
       "fast &= 1000\\\\\n",
       "\\frac{dy_p}{dt} &= z2_m*x_m+z2_p*x_p-fast*y_m*y_p\\\\\n",
       "\\frac{dy_m}{dt} &= z2_m*x_p+z2_p*x_m-fast*y_m*y_p\\\\\n",
       "\\frac{dz2_p}{dt} &= y_p*x_m+y_m*x_p-fast*z2_m*z2_p\\\\\n",
       "\\frac{dz2_m}{dt} &= y_m*x_m+y_p*x_p-fast*z2_m*z2_p\\\\\n",
       "\\frac{dx_p}{dt} &= x_m-fast*x_m*x_p\\\\\n",
       "\\frac{dx_m}{dt} &= x_p-fast*x_m*x_p\\\\\n",
       "\\end{align*}\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "list_ode."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerical_simulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div class=\"bk-root\">\n",
       "    <div class=\"bk-plotdiv\" id=\"4e0b256c-7062-4cd3-b5e6-aa5b54e79364\"></div>\n",
       "</div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
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       "  var render_items = [{\"docid\":\"711fa0d5-23fb-4fc9-902f-01722e53f4da\",\"elementid\":\"4e0b256c-7062-4cd3-b5e6-aa5b54e79364\",\"modelid\":\"4e720808-1d36-4e23-9ca4-510b9cc560dc\",\"notebook_comms_target\":\"47bdeb96-b7af-4bad-ab81-5c4661358bca\"}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "4e720808-1d36-4e23-9ca4-510b9cc560dc"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerical_simulation(method:ssa)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div class=\"bk-root\">\n",
       "    <div class=\"bk-plotdiv\" id=\"e74800ad-ad0f-4ccc-abf2-fb69ffa2ec04\"></div>\n",
       "</div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
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$y\"]]},\"id\":\"7f179d2e-63e0-4bae-9253-c563799f8603\",\"type\":\"HoverTool\"},{\"attributes\":{\"callback\":null},\"id\":\"77afc200-35f3-4ce4-9908-a214ac835748\",\"type\":\"DataRange1d\"},{\"attributes\":{},\"id\":\"c5855b0e-ecba-4f90-a110-631892b66f52\",\"type\":\"LinearScale\"},{\"attributes\":{\"formatter\":{\"id\":\"ccb533af-1ca0-487a-bd18-a8bf8fc5fd23\",\"type\":\"BasicTickFormatter\"},\"plot\":{\"id\":\"d1950695-365b-4a69-8996-550ca79098e5\",\"subtype\":\"Figure\",\"type\":\"Plot\"},\"ticker\":{\"id\":\"0dd18f3c-deae-4ee2-8107-8c80eb22d353\",\"type\":\"BasicTicker\"}},\"id\":\"43dec208-9b28-4db4-a69d-0afe05f98bff\",\"type\":\"LinearAxis\"},{\"attributes\":{\"line_alpha\":0.1,\"line_color\":\"#1f77b4\",\"line_width\":2,\"x\":{\"field\":\"Time\"},\"y\":{\"field\":\"x_m\"}},\"id\":\"bc821408-a15f-40af-92db-209893361d54\",\"type\":\"Line\"},{\"attributes\":{\"label\":{\"value\":\"y_p\"},\"renderers\":[{\"id\":\"ffef6d28-3b20-4a02-9d27-461db8e5b505\",\"type\":\"GlyphRenderer\"}]},\"id\":\"79c0695b-6d42-4b87-a25e-82a5115b1973\",\"type\":\"LegendItem\"},{\"attributes\":{\"line_alpha\":0.1,\"line_color\":\"#1f77b4\",\"line_width\":2,\"x\":{\"field\":\"Time\"},\"y\":{\"field\":\"z2_p\"}},\"id\":\"dbd048f1-9d94-499b-8d3b-11915f63e852\",\"type\":\"Line\"},{\"attributes\":{\"indices\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149]},\"id\":\"ad8a533b-6e5b-47dc-b827-519754b20733\",\"type\":\"IndexFilter\"},{\"attributes\":{\"line_color\":\"#d62728\",\"line_width\":2,\"x\":{\"field\":\"Time\"},\"y\":{\"field\":\"y_p\"}},\"id\":\"a8ff522b-4550-4d02-a565-7fa79c1e7f50\",\"type\":\"Line\"},{\"attributes\":{\"active_drag\":{\"id\":\"aee85722-76cb-422c-9a03-c6b3b656524a\",\"type\":\"BoxZoomTool\"},\"active_inspect\":\"auto\",\"active_scroll\":null,\"active_tap\":\"auto\",\"tools\":[{\"id\":\"1dd578b6-2fc1-415e-954c-7b09d07780cc\",\"type\":\"PanTool\"},{\"id\":\"2b6a05f9-271f-4ebd-9558-6b37ae2f42a5\",\"type\":\"WheelZoomTool\"},{\"id\":\"aee85722-76cb-422c-9a03-c6b3b656524a\",\"type\":\"BoxZoomTool\"},{\"id\":\"4bc03bf3-9938-40c3-a257-ed808ea294bb\",\"type\":\"ResetTool\"},{\"id\":\"30aa2867-89d7-4c1a-b4fc-74411d173cfd\",\"type\":\"SaveTool\"},{\"id\":\"d42a5a1a-7d37-437a-a844-2300e817db6a\",\"type\":\"UndoTool\"},{\"id\":\"7f179d2e-63e0-4bae-9253-c563799f8603\",\"type\":\"HoverTool\"}]},\"id\":\"75961078-abd2-4e87-9aae-a0611b946b48\",\"type\":\"Toolbar\"},{\"attributes\":{},\"id\":\"0dd18f3c-deae-4ee2-8107-8c80eb22d353\",\"type\":\"BasicTicker\"},{\"attributes\":{\"border_line_color\":{\"value\":\"gray\"},\"click_policy\":\"hide\",\"items\":[{\"id\":\"b56c21ec-ac6c-43cc-973c-e7a87d988652\",\"type\":\"LegendItem\"},{\"id\":\"d32934df-2415-43dc-9d33-6f2886abda2e\",\"type\":\"LegendItem\"},{\"id\":\"844604bd-e1eb-4399-9a88-81b7f0c2dd32\",\"type\":\"LegendItem\"},{\"id\":\"79c0695b-6d42-4b87-a25e-82a5115b1973\",\"type\":\"LegendItem\"},{\"id\":\"a891a1ea-3454-4954-b308-496f710a3552\",\"type\":\"LegendItem\"},{\"id\":\"9d3e44d2-bf7d-4464-903a-0febd2f3e203\",\"type\":\"LegendItem\"}],\"location\":[10,0],\"plot\":{\"id\":\"d1950695-365b-4a69-8996-550ca79098e5\",\"subtype\":\"Figure\",\"type\":\"Plot\"}},\"id\":\"ea47c046-2822-4926-8aca-7e808c024484\",\"type\":\"Legend\"},{\"attributes\":{\"line_color\":\"#1f77b4\",\"line_width\":2,\"x\":{\"field\":\"Time\"},\"y\":{\"field\":\"x_m\"}},\"id\":\"f825aaf3-7eee-4e1a-b54e-97468b2ba1d2\",\"type\":\"Line\"},{\"attributes\":{},\"id\":\"1dd578b6-2fc1-415e-954c-7b09d07780cc\",\"type\":\"PanTool\"},{\"attributes\":{\"data_source\":{\"id\":\"8fa3a32f-cecb-468b-93a0-592913e0180c\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"a8ff522b-4550-4d02-a565-7fa79c1e7f50\",\"type\":\"Line\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"11e38f54-ebad-44fc-997e-ada802e8a09b\",\"type\":\"Line\"},\"selection_glyph\":null,\"view\":{\"id\":\"38f93dd8-7125-4468-9e04-4f92b96049f7\",\"type\":\"CDSView\"}},\"id\":\"ffef6d28-3b20-4a02-9d27-461db8e5b505\",\"type\":\"GlyphRenderer\"},{\"attributes\":{},\"id\":\"0cbae3b5-e9c4-42e2-81ab-64ca66fa1456\",\"type\":\"LinearScale\"}],\"root_ids\":[\"d1950695-365b-4a69-8996-550ca79098e5\"]},\"title\":\"Bokeh Application\",\"version\":\"0.12.14\"}};\n",
       "  var render_items = [{\"docid\":\"028500a4-68d0-4575-9b45-e89bce6de76c\",\"elementid\":\"e74800ad-ad0f-4ccc-abf2-fb69ffa2ec04\",\"modelid\":\"d1950695-365b-4a69-8996-550ca79098e5\",\"notebook_comms_target\":\"00563e43-9746-4736-8394-3f795fb39426\"}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "d1950695-365b-4a69-8996-550ca79098e5"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sigmoid functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "clear_model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Logistic function 1/(1+e^-t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "S => 2*S."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "S =[S]=> _."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "present(S,0.5)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align*}\n",
       "S_0 &= 0.5\\\\\n",
       "\\frac{dS}{dt} &= S-S^2\\\\\n",
       "\\end{align*}\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "list_ode."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Hyperbolic tangent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "_ => HT."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "2*HT => HT."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align*}\n",
       "HT_0 &= 0\\\\\n",
       "S_0 &= 0.5\\\\\n",
       "\\frac{dHT}{dt} &= 1-HT^2\\\\\n",
       "\\frac{dS}{dt} &= S-S^2\\\\\n",
       "\\end{align*}\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "list_ode."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Arc tangent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "_ => T."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "1/(1+T^2) for /T => AT."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align*}\n",
       "AT_0 &= 0\\\\\n",
       "/T_0 &= 0\\\\\n",
       "T_0 &= 0\\\\\n",
       "HT_0 &= 0\\\\\n",
       "S_0 &= 0.5\\\\\n",
       "\\frac{dAT}{dt} &= 1/(1+T^2)\\\\\n",
       "\\frac{d/T}{dt} &= - (1/(1+T^2))\\\\\n",
       "\\frac{dT}{dt} &= 1\\\\\n",
       "\\frac{dHT}{dt} &= 1-HT^2\\\\\n",
       "\\frac{dS}{dt} &= S-S^2\\\\\n",
       "\\end{align*}\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "list_ode."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Hill function degree 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "MA(1) for NH1 =[NH1]=> _."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "MA(1) for _ =[2*NH1]=> H1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "present(NH1,1)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Hill function degree 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "MA(2) for NH2 =[T+NH2]=> _."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "MA(2) for _ =[T+2*NH2]=> H2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "present(NH2,1)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Hill function degree 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "MA(5) for NH5 =[4*T+NH5]=> _."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "MA(5) for _ =[4*T+2*NH5]=> H5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "present(NH5,1)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Hill function degree n\n",
    " Hn=t^n/(k+t^n)\n",
    "\n",
    " MA(n) for NHn =[(n-1)*t+NHn]=> _.\n",
    "\n",
    " MA(n*k) for _ =[(n-1)*t+2*NHn]=> Hn.\n",
    "\n",
    " present(NHn,1/k)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "_=[S]=>S.\r\n",
       "S=[S]=>_.\r\n",
       "_=>HT.\r\n",
       "HT=[HT]=>_.\r\n",
       "_=>T.\r\n",
       "1/(1+T^2)for/T=>AT.\r\n",
       "NH1=[NH1]=>_.\r\n",
       "_=[2*NH1]=>H1.\r\n",
       "MA(2)for NH2=[T+NH2]=>_.\r\n",
       "MA(2)for _=[T+2*NH2]=>H2.\r\n",
       "MA(5)for NH5=[4*T+NH5]=>_.\r\n",
       "MA(5)for _=[4*T+2*NH5]=>H5.\r\n",
       "present(S,0.5).\r\n",
       "present(NH1,1).\r\n",
       "present(NH2,1).\r\n",
       "present(NH5,1).\r\n"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list_model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align*}\n",
       "H5_0 &= 0\\\\\n",
       "NH5_0 &= 1\\\\\n",
       "H2_0 &= 0\\\\\n",
       "NH2_0 &= 1\\\\\n",
       "H1_0 &= 0\\\\\n",
       "NH1_0 &= 1\\\\\n",
       "AT_0 &= 0\\\\\n",
       "/T_0 &= 0\\\\\n",
       "T_0 &= 0\\\\\n",
       "HT_0 &= 0\\\\\n",
       "S_0 &= 0.5\\\\\n",
       "\\frac{dH5}{dt} &= 5*T^4*NH5^2\\\\\n",
       "\\frac{dNH5}{dt} &= - (5*T^4*NH5^2)\\\\\n",
       "\\frac{dH2}{dt} &= 2*T*NH2^2\\\\\n",
       "\\frac{dNH2}{dt} &= - (2*T*NH2^2)\\\\\n",
       "\\frac{dH1}{dt} &= NH1^2\\\\\n",
       "\\frac{dNH1}{dt} &= -NH1^2\\\\\n",
       "\\frac{dAT}{dt} &= 1/(1+T^2)\\\\\n",
       "\\frac{d/T}{dt} &= - (1/(1+T^2))\\\\\n",
       "\\frac{dT}{dt} &= 1\\\\\n",
       "\\frac{dHT}{dt} &= 1-HT^2\\\\\n",
       "\\frac{dS}{dt} &= S-S^2\\\\\n",
       "\\end{align*}\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "list_ode."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerical_simulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div class=\"bk-root\">\n",
       "    <div class=\"bk-plotdiv\" id=\"78842eed-4feb-4499-a5ec-4ef307da661c\"></div>\n",
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Application\",\"version\":\"0.12.14\"}};\n",
       "  var render_items = [{\"docid\":\"a3768726-14fe-43cf-a782-259c059dfa1d\",\"elementid\":\"78842eed-4feb-4499-a5ec-4ef307da661c\",\"modelid\":\"80673db9-9cc2-4470-b4d3-e36f9444764a\",\"notebook_comms_target\":\"dac8111f-8473-428b-866b-325f68c96961\"}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "80673db9-9cc2-4470-b4d3-e36f9444764a"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(show:{S,HT,AT,H1,H2,H5})."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerical_simulation(method:ssa)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div class=\"bk-root\">\n",
       "    <div class=\"bk-plotdiv\" id=\"b1c54372-7672-40dc-b69f-81b407c14119\"></div>\n",
       "</div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
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       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "4cec2b7c-4b6b-4080-acf6-6cb6e2b76377"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(show:{S,HT,AT,H1,H2,H5})."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Biocham",
   "language": "",
   "name": "biocham"
  },
  "language_info": {
   "codemirror_mode": "biocham",
   "file_extension": ".bc",
   "mimetype": "text/plain",
   "name": "biocham",
   "pygments_lexer": "prolog"
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 },
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