diff --git a/Esiur.Analysis.Test/Program.cs b/Esiur.Analysis.Test/Program.cs
index 39f078b..bb05509 100644
--- a/Esiur.Analysis.Test/Program.cs
+++ b/Esiur.Analysis.Test/Program.cs
@@ -39,6 +39,9 @@ namespace Esiur.Analysis.Test
         [STAThread]
         static void Main()
         {
+
+            var f = Esiur.Analysis.Algebra.Functions.Sigmoid;
+
             var signalA = new double[] { V,1, V, 1 , V, V, V };
             var signalB = new double[] { V, V, 1, V, V, 1, V };
              var cor = signalA.CrossCorrelation(signalB, true);
diff --git a/Esiur.Analysis/Algebra/Functions.cs b/Esiur.Analysis/Algebra/Functions.cs
new file mode 100644
index 0000000..e8f1cfa
--- /dev/null
+++ b/Esiur.Analysis/Algebra/Functions.cs
@@ -0,0 +1,42 @@
+using Esiur.Analysis.Neural;
+using System;
+using System.Collections.Generic;
+using System.Text;
+
+namespace Esiur.Analysis.Algebra
+{
+
+    public static class Functions
+    {
+        private static MathFunction<RealFunction> SigmoidDerivative
+            = new MathFunction<RealFunction>(new RealFunction(x => x * (1 - x)), null, null);
+
+        public static MathFunction<RealFunction> Sigmoid = new MathFunction<RealFunction>(new RealFunction(x =>
+        {
+            double k = Math.Exp(x);
+            return k / (1.0 + k);
+        }), SigmoidDerivative, null);
+
+
+        private static MathFunction<RealFunction> TanhDerivative = new MathFunction<RealFunction>(
+                new RealFunction(x => 1 - (x * x)), null, null);
+
+        public static MathFunction<RealFunction> Tanh = new MathFunction<RealFunction>(
+            new RealFunction(x => Math.Tanh(x)), TanhDerivative, null);
+
+
+        private static MathFunction<RealFunction> ReLUDerivative = new MathFunction<RealFunction>(
+        new RealFunction(x => (0 >= x) ? 0 : 1), null, null);
+
+        public static MathFunction<RealFunction> ReLU = new MathFunction<RealFunction>(
+            new RealFunction(x => (0 >= x) ? 0 : x), ReLUDerivative, null);
+
+
+        private static MathFunction<RealFunction> LeakyReLUDerivative = new MathFunction<RealFunction>(
+            new RealFunction(x => (0 >= x) ? 0.01 : 1), null, null);
+
+        public static MathFunction<RealFunction> LeakyReLU = new MathFunction<RealFunction>(
+            new RealFunction(x => (0 >= x) ? 0.01 * x : x), LeakyReLUDerivative, null);
+
+    }
+}
\ No newline at end of file
diff --git a/Esiur.Analysis/Algebra/MathFunction.cs b/Esiur.Analysis/Algebra/MathFunction.cs
new file mode 100644
index 0000000..f1113f6
--- /dev/null
+++ b/Esiur.Analysis/Algebra/MathFunction.cs
@@ -0,0 +1,29 @@
+using Esiur.Analysis.Neural;
+using System;
+using System.Collections.Generic;
+using System.Text;
+
+namespace Esiur.Analysis.Algebra
+{
+
+    public delegate double RealFunction(double x);
+    public delegate double Real2Function(double x, double y);
+    public delegate double Real3Function(double x, double y, double z);
+
+    public class MathFunction<T>
+    {
+        public T Function { get; internal set; }
+        public MathFunction<T> Derivative { get; internal set; }
+        public MathFunction<T> Integral { get; internal set; }
+
+        public MathFunction(T function, MathFunction<T> derivative, MathFunction<T> integral)
+        {
+            Function = function;
+            Derivative = derivative;
+            Integral = integral;
+        }
+
+     }
+
+
+}
diff --git a/Esiur.Analysis/Neural/ActivationFunction.cs b/Esiur.Analysis/Neural/ActivationFunction.cs
new file mode 100644
index 0000000..b76c4c8
--- /dev/null
+++ b/Esiur.Analysis/Neural/ActivationFunction.cs
@@ -0,0 +1,87 @@
+using System;
+using System.Collections.Generic;
+using System.Text;
+
+namespace Esiur.Analysis.Neural
+{
+    public delegate double ActivationFunction(double value);
+
+    public static class ActivationFunctions
+    {
+
+        public static (ActivationFunction, ActivationFunction) Sigmoid()
+        {
+            return (new ActivationFunction(x =>
+            {
+                float k = (float)Math.Exp(x);
+                return k / (1.0 + k);
+            }),
+              new ActivationFunction(x =>
+             {
+                 return x * (1 - x);
+             }));
+        }
+
+        public static ActivationFunction Tanh()
+        {
+            return new ActivationFunction(x =>
+            {
+                return Math.Tanh(x);
+            });
+        }
+
+
+        public static ActivationFunction ReLU()
+        {
+            return new ActivationFunction(x =>
+            {
+                return (0 >= x) ? 0 : x;
+            });
+        }
+
+        public static ActivationFunction LeakyReLU()
+        {
+            return new ActivationFunction(x =>
+            {
+                return (0 >= x) ? 0.01 * x : x;
+            });
+        }
+
+        public static ActivationFunction SigmoidDer()
+        {
+            return new ActivationFunction(x =>
+            {
+                return x * (1 - x);
+            });
+        }
+
+        public static ActivationFunction TanhDer()
+        {
+            return new ActivationFunction(x =>
+            {
+                return 1 - (x * x);
+            });
+        }
+
+        public static ActivationFunction ReLUDer()
+        {
+            return new ActivationFunction(x =>
+            {
+                return (0 >= x) ? 0 : 1;
+            });
+        }
+
+        public static ActivationFunction LeakyReLUDer()
+        {
+            return new ActivationFunction(x =>
+            {
+                return (0 >= x) ? 0.01f : 1;
+            });
+        }
+
+
+
+        
+    }
+
+}
diff --git a/Esiur.Analysis/Neural/NeuralLayer.cs b/Esiur.Analysis/Neural/NeuralLayer.cs
new file mode 100644
index 0000000..5fe8cdc
--- /dev/null
+++ b/Esiur.Analysis/Neural/NeuralLayer.cs
@@ -0,0 +1,50 @@
+using Esiur.Analysis.Algebra;
+using Microsoft.CodeAnalysis;
+using System;
+using System.Collections.Generic;
+using System.Text;
+
+namespace Esiur.Analysis.Neural
+{
+    internal class NeuralLayer
+    {
+
+
+        public Neuron[] Neurons { get; internal set; }
+
+        public MathFunction<RealFunction> Activation { get; internal set; }
+
+        public NeuralLayer? PreviousLayer { get; internal set; }
+        public double Bias { get; set; }
+
+        public NeuralLayer(int nodes, MathFunction<RealFunction> activation, NeuralLayer? previousLayer)
+        {
+
+            PreviousLayer = previousLayer;
+            Neurons = new Neuron[nodes];
+            Activation = activation;
+
+            for(var i = 0; i < nodes; i++)
+            {
+                var synapses = new List<Synapse>();
+
+                var neuron = new Neuron()
+                {
+                    Layer = this,
+                };
+
+
+                if (previousLayer != null)
+                {
+                    for(var j = 0; j < previousLayer.Neurons.Length; j++)
+                    {
+                        synapses.Add(new Synapse() { Source = previousLayer.Neurons[j] , Target = neuron, Weight = 0});
+                    }
+                }
+
+                Neurons[i] = neuron;   
+            }
+
+        }
+    }
+}
diff --git a/Esiur.Analysis/Neural/NeuralNetwork.cs b/Esiur.Analysis/Neural/NeuralNetwork.cs
new file mode 100644
index 0000000..9ce1c0e
--- /dev/null
+++ b/Esiur.Analysis/Neural/NeuralNetwork.cs
@@ -0,0 +1,101 @@
+using Esiur.Analysis.Algebra;
+using System;
+using System.Collections.Generic;
+using System.Linq;
+using System.Text;
+
+namespace Esiur.Analysis.Neural
+{
+    public class NeuralNetwork
+    {
+        NeuralLayer[] neuralLayers;
+
+        public NeuralNetwork(int[] layers, MathFunction<RealFunction>[] activations)
+        {
+            neuralLayers = new NeuralLayer[layers.Length];
+
+            for (var i = 0; i < layers.Length; i++)
+                neuralLayers[i] = new NeuralLayer(layers[i], activations[i], i == 0 ? null : neuralLayers[i-1]);
+        }
+
+        public double[] FeedForward(double[] input)
+        {
+            for (int i = 0; i < input.Length; i++)
+            {
+                neuralLayers[0].Neurons[i].Value = input[i];
+            }
+
+            for (int i = 1; i < neuralLayers.Length; i++)
+            {
+                for (int j = 0; j < neuralLayers[i].Neurons.Length; j++)
+                {
+                    neuralLayers[i].Neurons[j].Forward();
+                }
+            }
+
+            return neuralLayers.Last().Neurons.Select(x => x.Value).ToArray();
+        }
+
+
+
+        public void BackPropagate(double[] input, double[] target)
+        {
+            var output = FeedForward(input);
+
+            // total error (square error function)
+            double totalError = 0.5 * output.Zip(target, (x, y) => Math.Pow(x - y, 2)).Sum() ;
+
+
+            // calculate partial derivitave of E-Total with respect to weights dE/dW = dE/dOF * dOF/dO *  dO/dW
+
+           for(var i = 0; i < target.Length; i++)
+            {
+                var z = -(target[i] - output[i]) * 
+            }
+            //for (int i = 0; i < output.Length; i++) 
+            //    totalError += (float)Math.Pow(output[i] - expected[i], 2);//calculated cost of network
+
+            //totalError /= 2; //this value is not used in calculions, rather used to identify the performance of the network
+
+
+
+            var gamma = neuralLayers.Select(x => x.Neurons.Select(n => n.Value).ToArray()).ToArray();
+
+ 
+            int layer = layers.Length - 2;
+
+            for (int i = 0; i < output.Length; i++) gamma[layers.Length - 1][i] = (output[i] - expected[i]) * activateDer(output[i], layer);//Gamma calculation
+            for (int i = 0; i < layers[layers.Length - 1]; i++)//calculates the w' and b' for the last layer in the network
+            {
+                biases[layers.Length - 2][i] -= gamma[layers.Length - 1][i] * learningRate;
+                for (int j = 0; j < layers[layers.Length - 2]; j++)
+                {
+
+                    weights[layers.Length - 2][i][j] -= gamma[layers.Length - 1][i] * neurons[layers.Length - 2][j] * learningRate;//*learning 
+                }
+            }
+
+            for (int i = layers.Length - 2; i > 0; i--)//runs on all hidden layers
+            {
+                layer = i - 1;
+                for (int j = 0; j < layers[i]; j++)//outputs
+                {
+                    gamma[i][j] = 0;
+                    for (int k = 0; k < gamma[i + 1].Length; k++)
+                    {
+                        gamma[i][j] += gamma[i + 1][k] * weights[i][k][j];
+                    }
+                    gamma[i][j] *= activateDer(neurons[i][j], layer);//calculate gamma
+                }
+                for (int j = 0; j < layers[i]; j++)//itterate over outputs of layer
+                {
+                    biases[i - 1][j] -= gamma[i][j] * learningRate;//modify biases of network
+                    for (int k = 0; k < layers[i - 1]; k++)//itterate over inputs to layer
+                    {
+                        weights[i - 1][j][k] -= gamma[i][j] * neurons[i - 1][k] * learningRate;//modify weights of network
+                    }
+                }
+            }
+        }
+    }
+}
diff --git a/Esiur.Analysis/Neural/Neuron.cs b/Esiur.Analysis/Neural/Neuron.cs
new file mode 100644
index 0000000..5b37cc4
--- /dev/null
+++ b/Esiur.Analysis/Neural/Neuron.cs
@@ -0,0 +1,21 @@
+using System;
+using System.Collections.Generic;
+using System.Linq;
+using System.Text;
+
+namespace Esiur.Analysis.Neural
+{
+    internal class Neuron
+    {
+        public double Value { get; set; }
+        public NeuralLayer Layer { get; set; }
+
+        public List<Synapse> Synapses { get; set; } = new List<Synapse>();
+
+        public void Forward()
+        {
+            var sum = Synapses.Sum(x => x.Weight * x.Source.Value);
+            Value = Layer.Activation.Function(sum + Layer.PreviousLayer?.Bias ?? 0);
+        }
+    }
+}
diff --git a/Esiur.Analysis/Neural/Synapse.cs b/Esiur.Analysis/Neural/Synapse.cs
new file mode 100644
index 0000000..a874917
--- /dev/null
+++ b/Esiur.Analysis/Neural/Synapse.cs
@@ -0,0 +1,14 @@
+using System;
+using System.Collections.Generic;
+using System.Text;
+
+namespace Esiur.Analysis.Neural
+{
+    internal class Synapse
+    {
+        public double Weight { get; set; }
+
+        public Neuron Source { get; set; }
+        public Neuron Target { get; set; }
+    }
+}