discrete time steps earlier. Problem-specific LSTM-like topologies can be evolved. LSTM works even given long delays between significant events and can handle signals that mix low and high frequency components. Many applications use stacks of LSTM RNNs and train them by Connectionist Temporal Classification (CTC) to find an RNN weight matrix that maximizes the probability of the label sequences in a training set, given the corresponding input sequences. CTC achieves both alignment and recognition. LSTM can learn to recognize context-sensitive languages unlike previous models based on hidden Markov models (HMM) and similar concepts. Gated recurrent unit Gated recurrent units (GRUs) are a gating mechanism in recurrent neural networks introduced in 2014. They are used in the full form and several simplified variants. Their performance on polyphonic music modeling and speech signal modeling was found to be similar to that of long short-term memory. They have fewer parameters than LSTM, as they lack an output gate. Bi-directional Bi-directional RNNs use a finite sequence to predict or label each element of the sequence based on the element's past and future contexts. This is done by concatenating the outputs of two RNNs, one processing the sequence from left to right, the other one from right to left. The combined outputs are the predictions of the teacher-given target signals. This technique has been proven to be especially useful when combined with LSTM RNNs. Continuous-time A continuous-time recurrent neural network (CTRNN) uses a system of ordinary differential equations to model the effects on a neuron of the incoming inputs. For a neuron in the network with activation , the rate of change of activation is given by: Where: : Time constant of postsynaptic node : Activation of postsynaptic node : Rate of change of activation of postsynaptic node : Weight of connection from pre to postsynaptic node : Sigmoid of x e.g. . : Activation of presynaptic node : Bias of presynaptic node : Input (if any) to node CTRNNs have been applied to evolutionary robotics where they have been used to address vision, co-operation, and minimal cognitive behaviour. Note that, by the Shannon sampling theorem, discrete time recurrent neural networks can be viewed as continuous-time recurrent neural networks where the differential equations have transformed into equivalent difference equations. This transformation can be thought of as occurring after the post-synaptic node activation functions have been low-pass filtered but prior to sampling. Hierarchical recurrent neural network